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Since Beck, Katz, and Tucker (1998), the standard method for modeling time dependence in binary data has been to incorporate time dummies or splined time in logistic regressions. Although we agree with the need for modeling time dependence, we demonstrate that time dummies can induce estimation problems due to separation. Splines do not suffer from these problems. However, the complexity of splines has led substantive researchers (1) to use knot values that may be inappropriate for their data and (2) to ignore any substantive discussion concerning temporal dependence. We propose a relatively simple alternative: including t, t 2, and t 3 in the regression. This cubic polynomial approximation is trivial to implement—and, therefore, interpret—and it avoids problems such as quasi-complete separation. Monte Carlo analysis demonstrates that, for the types of hazards one often sees in substantive research, the polynomial approximation always outperforms time dummies and generally performs as well as splines or even more flexible autosmoothing procedures. Due to its simplicity, this method also accommodates nonproportional hazards in a straightforward way. We reanalyze Crowley and Skocpol (2001) using nonproportional hazards and find new empirical support for the historical-institutionalist perspective.

In this paper, we prove the assertion that the number of monic cubic polynomials $F(x) = x^3 + a_2 x^2 + a_1 x + a_0$ with integer coefficients and irreducible, Galois over ${\\mathbb {Q}}$ satisfying $\\max \\{|a_2|, |a_1|, |a_0|\\} \\leq X$ is bounded from above by $O(X (\\log X)^2)$. We also count the number of abelian monic binary cubic forms with integer coefficients up to a natural equivalence relation ordered by the so-called Bhargava–Shankar height. Finally, we prove an assertion characterizing the splitting field of 2-torsion points of semi-stable abelian elliptic curves.

This study utilizes regression discontinuity to examine the long-run impacts of the mita, an extensive forced mining labor system in effect in Peru and Bolivia between 1573 and 1812. Results indicate that a mita effect lowers household consumption by around 25% and increases the prevalence of stunted growth in children by around 6 percentage points in subjected districts today. Using data from the Spanish Empire and Peruvian Republic to trace channels of institutional persistence, I show that the mita's influence has persisted through its impacts on land tenure and public goods provision. Mita districts historically had fewer large landowners and lower educational attainment. Today, they are less integrated into road networks and their residents are substantially more likely to be subsistence farmers.

We are concerned with the intersections between the boundaries of bounded hyperbolic components in the space of cubic polynomials. We show that if the intersection map has an attracting periodic orbit, then it is either a parabolic map or a Misiurewicz map. This result is based on a complete classification of all these intersection maps in one-dimensional slices S p introduced by J. Milnor, which consists of cubic polynomials with a periodic p critical point.

Self-determination theory (SDT) proposes to explain the relations between motivational states and human development. In education, a central tenet of the theory is that experiencing autonomous motivation in school activities (i.e., genuine pleasure and enjoyment) fosters optimal learning processes, whereas experiencing controlled motivation (i.e., pressure from social or instrumental incentives) undermines them. Although the theory is well established empirically, little is known about how these motivations combine in their effects on achievement at school (interactions), and whether their effects depend on the intensity of motivations (nonlinearities) or on the context of study (national differences). Applying cubic response surface analysis to the TIMSS 2019 dataset on mathematics (N = 152,825 8th grade students from 37 countries), as well as replication data (N = 169,269 8th grade students from TIMSS 2015, N = 270 college students from SDT data), this study uncovers the existence of various kinds of nonlinear-interactive motivational processes in achievement, three of which systematically account for cross-national differences. In substance, these findings demonstrate that predictions based on SDT are close to universally true (93% of students), although they may not generalize well to extreme states of autonomous or controlled motivation (nonlinear and interactive processes). Implications for research and interventions on motivational processes are discussed.

In this paper, we propose to present a novel technique for designing cryptographically strong substitution-boxes using cubic polynomial mapping. The proposed cubic polynomial mapping is proficient to map the input sequence to a strong 8 × 8 S-box meeting the requirements of a bijective function. The use of cubic polynomial maintains the simplicity of S-box construction method and found consistent when compared with other existing S-box techniques used to construct S-boxes. An example proposed S-box is obtained which is analytically evaluated using standard performance criteria including nonlinearity, bijection, bit independence, strict avalanche effect, linear approximation probability, and differential uniformity. The performance results are equated with some recently scrutinized S-boxes to ascertain its cryptographic forte. The critical analyses endorse that the proposed S-box construction technique is considerably innovative and effective to generate cryptographic strong substitution-boxes.

Recently, facial-expression recognition (FER) has primarily focused on images in the wild, including factors such as face occlusion and image blurring, rather than laboratory images. Complex field environments have introduced new challenges to FER. To address these challenges, this study proposes a cross-fusion dual-attention network. The network comprises three parts: (1) a cross-fusion grouped dual-attention mechanism to refine local features and obtain global information; (2) a proposed C 2 activation function construction method, which is a piecewise cubic polynomial with three degrees of freedom, requiring less computation with improved flexibility and recognition abilities, which can better address slow running speeds and neuron inactivation problems; and (3) a closed-loop operation between the self-attention distillation process and residual connections to suppress redundant information and improve the generalization ability of the model. The recognition accuracies on the RAF-DB, FERPlus, and AffectNet datasets were 92.78%, 92.02%, and 63.58%, respectively. Experiments show that this model can provide more effective solutions for FER tasks.

Let P 3 ( x , y ) and Q 3 ( x , y ) be polynomials of degree three without constant or linear terms. We characterize the global centers of all polynomial differential systems of the form x ˙ = y + P 3 ( x , y ) , y ˙ = Q 3 ( x , y ) that are reversible and invariant with respect to the x -axis.

The rapid acceleration of urbanization and the surge in car ownership necessitate efficient automatic parking solutions in constricted spaces to address the escalating urban parking issue. To optimize space utilization, enhance traffic efficiency, and mitigate accident risks, a method is proposed for smooth, comfortable, and adaptable automatic parking trajectory planning. This study initially employs a hybrid A* algorithm to generate a preliminary path, then fits the velocity and acceleration based on a cubic polynomial. The kinematic constraints of the vehicle and obstacle avoidance constraints are then meticulously defined, and a coupled nonlinear model predictive control (NMPC) method is employed to optimize the trajectory. Compared to the hybrid A* algorithm, the optimized trajectory demonstrates superior space utilization and improved smoothness. The experimental results indicate that the proposed method performs effectively in automated parking tasks in confined spaces, suggesting promising applications and broad prospects for future.

A DEA-based stochastic frontier estimation framework is presented to evaluate contextual variables affecting productivity that allows for both one-sided inefficiency deviations as well as two-sided random noise. Conditions are identified under which a two-stage procedure consisting of DEA followed by ordinary least squares (OLS) regression analysis yields consistent estimators of the impact of contextual variables. Conditions are also identified under which DEA in the first stage followed by maximum likelihood estimation (MLE) in the second stage yields consistent estimators of the impact of contextual variables. This requires the contextual variables to be independent of the input variables, but the contextual variables may be correlated with each other. Monte Carlo simulations are carried out to compare the performance of our two-stage approach with one-stage and two-stage parametric approaches. Simulation results indicate that DEA-based procedures with OLS, maximum likelihood, or even Tobit estimation in the second stage perform as well as the best of the parametric methods in the estimation of the impact of contextual variables on productivity. Simulation results also indicate that DEA-based procedures perform better than parametric methods in the estimation of individual decision-making unit (DMU) productivity. Overall, the results establish DEA as a nonparametric stochastic frontier estimation (SFE) methodology.