From the Law of Value Debate to the One-Child Policy in China: On Accounting and Biopolitics

This article argues that the origins of the one-child policy beginning in 1980 in China, and its development into the current system of “comprehensive population management,” are to be found not in any unfolding of a statist or authoritarian logic, or within the parameters of a nominally “socialist” project, but rather in a return to a properly capitalist set of concerns and governmental techniques, the first iteration of which can be traced to the 1920s and 1930s. With regard to the broad set of economic reforms launched in the period 1979–81, it is argued that the one-child policy is absolutely continuous with other reforms across economic sectors (agricultural responsibility systems and urban enterprise reforms) and discontinuous with anything we might understand as population management in the period 1949–76. The “law of value debate” in 1979, which “resolved” a long-standing set of issues concerning national accounting, planning, and accumulation, is found to be—despite its apparently Marxist character, derivation, and vocabulary—the passage through which a capitalist developmental logic was reintroduced into Chinese governing, with significant consequences.

Relaxation of functionals with linear growth: Interactions of emerging measures and free discontinuities

For an integral functional defined on functions ( u , v ) ∈ W 1 , 1 × L 1 {(u,v)\in W^{1,1}\times L^{1}} featuring a prototypical strong interaction term between u and v, we calculate its relaxation in the space of functions with bounded variations and Radon measures. Interplay between measures and discontinuities brings various additional difficulties, and concentration effects in recovery sequences play a major role for the relaxed functional even if the limit measures are absolutely continuous with respect to the Lebesgue one.

Fuglede’s theorem in generalized Orlicz–Sobolev spaces

In this paper, we show that Orlicz–Sobolev spaces $$W^{1,\varphi }(\varOmega )$$ W 1 , φ ( Ω ) can be characterized with the ACL- and ACC-characterizations. ACL stands for absolutely continuous on lines and ACC for absolutely continuous on curves. Our results hold under the assumptions that $$C^1(\varOmega )$$ C 1 ( Ω ) functions are dense in $$W^{1,\varphi }(\varOmega )$$ W 1 , φ ( Ω ) , and $$\varphi (x,\beta ) \ge 1$$ φ ( x , β ) ≥ 1 for some $$\beta > 0$$ β > 0 and almost every $$x \in \varOmega $$ x ∈ Ω . The results are new even in the special cases of Orlicz and double phase growth.

Invariant measures for random expanding on average Saussol maps

In this paper, we investigate the existence of random absolutely continuous invariant measures (ACIP) for random expanding on average Saussol maps in higher dimensions. This is done by the establishment of a random Lasota–Yorke inequality for the transfer operators on the space of bounded oscillation. We prove that the number of ergodic skew product ACIPs is finite and will provide an upper bound for the number of these ergodic ACIPs. This work can be seen as a generalization of the work in [F. Batayneh and C. González-Tokman, On the number of invariant measures for random expanding maps in higher dimensions, Discrete Contin. Dyn. Syst. 41 (2021) 5887–5914] on admissible random Jabłoński maps to a more general class of higher-dimensional random maps.

Поточечное условие абсолютной непрерывности функции одной переменной и его применения

An absolutely continuous function in calculus is precisely such a function that, within the framework of Lebesgue integration, can be restored from its derivative, that is, the Newton--Leibniz theorem on the relationship between integration and differentiation is fulfilled for it. An equivalent definition is that the the sum of the moduli of the increments of the function with respect to arbitrary pair-wise disjoint intervals is less than any positive number if the sum of the lengths of the intervals is small enough. Certain sufficient conditions for absolute continuity are known, for example, the Banach--Zaretsky theorem. In this paper we prove a new sufficient condition for the absolute continuity of a function of one variable and give some of its applications to problems in the theory of function spaces. The proved condition makes it possible to significantly simplify the proof of the theorems on the pointwise description of functions of the Sobolev classes defined on Euclidean spaces and Сarnot groups.

Dynamical behavior of alternate base expansions

We generalize the greedy and lazy $\beta $ -transformations for a real base $\beta $ to the setting of alternate bases ${\boldsymbol {\beta }}=(\beta _0,\ldots ,\beta _{p-1})$ , which were recently introduced by the first and second authors as a particular case of Cantor bases. As in the real base case, these new transformations, denoted $T_{{\boldsymbol {\beta }}}$ and $L_{{\boldsymbol {\beta }}}$ respectively, can be iterated in order to generate the digits of the greedy and lazy ${\boldsymbol {\beta }}$ -expansions of real numbers. The aim of this paper is to describe the measure-theoretical dynamical behaviors of $T_{{\boldsymbol {\beta }}}$ and $L_{{\boldsymbol {\beta }}}$ . We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the p-Lebesgue measure) $T_{{\boldsymbol {\beta }}}$ -invariant measure. We then show that this unique measure is in fact equivalent to the p-Lebesgue measure and that the corresponding dynamical system is ergodic and has entropy $({1}/{p})\log (\beta _{p-1}\cdots \beta _0)$ . We give an explicit expression of the density function of this invariant measure and compute the frequencies of letters in the greedy ${\boldsymbol {\beta }}$ -expansions. The dynamical properties of $L_{{\boldsymbol {\beta }}}$ are obtained by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the $\beta $ -shift. Finally, we show that the ${\boldsymbol {\beta }}$ -expansions can be seen as $(\beta _{p-1}\cdots \beta _0)$ -representations over general digit sets and we compare both frameworks.

Absolutely continuous and pure point spectra of discrete operators with sparse potentials

We consider the discrete Schr\”odinger operator $H=-\Delta+V$ with a sparse potential $V$ and find conditions guaranteeing either existence of wave operators for the pair $H$ and $H_0=-\Delta$, or presence of dense purely point spectrum of the operator $H$ on some interval $[\lambda_0,0]$ with $\lambda_0<0$.

On Two Measure-Theoretic Aspects of the Full Bayesian Significance Test for Precise Bayesian Hypothesis Testing †

The Full Bayesian Significance Test (FBST) has been proposed as a convenient method to replace frequentist p-values for testing a precise hypothesis. Although the FBST enjoys various appealing properties, the purpose of this paper is to investigate two aspects of the FBST which are sometimes observed as measure-theoretic inconsistencies of the procedure and have not been discussed rigorously in the literature. First, the FBST uses the posterior density as a reference for judging the Bayesian statistical evidence against a precise hypothesis. However, under absolutely continuous prior distributions, the posterior density is defined only up to Lebesgue null sets which renders the reference criterion arbitrary. Second, the FBST statistical evidence seems to have no valid prior probability. It is shown that the former aspect can be circumvented by fixing a version of the posterior density before using the FBST, and the latter aspect is based on its measure-theoretic premises. An illustrative example demonstrates the two aspects and their solution. Together, the results in this paper show that both of the two aspects which are sometimes observed as measure-theoretic inconsistencies of the FBST are not tenable. The FBST thus provides a measure-theoretically coherent Bayesian alternative for testing a precise hypothesis.