In Arabidopsis thaliana, LESION SIMULATING DISEASE 1 (LSD1), ENHANCED DISEASE SUSCEPTIBILITY 1 (EDS1) and PHYTOALEXIN DEFICIENT 4 (PAD4) proteins are regulators of cell death (CD) in response to abiotic and biotic stresses. Hormones, such as salicylic acid (SA), and reactive oxygen species, such as hydrogen peroxide (H2O2), are key signaling molecules involved in plant CD. The proposed mathematical models presented in this study suggest that LSD1, EDS1 and PAD4 together with SA and H2O2 are involved in the control of plant water use efficiency (WUE), vegetative growth and generative development. The analysis of Arabidopsis wild‐type and single mutants lsd1, eds1, and pad4, as well as double mutants eds1/lsd1 and pad4/lsd1, demonstrated the strong conditional correlation between SA/H2O2 and WUE that is dependent on LSD1, EDS1 and PAD4 proteins. Moreover, we found a strong correlation between the SA/H2O2 homeostasis of 4‐week‐old Arabidopsis leaves and a total seed yield of 9‐week‐old plants. Altogether, our results prove that SA and H2O2 are conditionally regulated by LSD1/EDS/PAD4 to govern WUE, biomass accumulation and seed yield. Conditional correlation and the proposed models presented in this study can be used as the starting points in the creation of a plant breeding algorithm that would allow to estimate the seed yield at the initial stage of plant growth, based on WUE, SA and H2O2 content.

Chaundy and Jolliffe proved that if {ck}k=1∞ is a nonincreasing real sequence with limk→∞ck=0, then the series ∑k=1∞‍cksin⁡kx converges uniformly if and only if kck→0. The purpose of this paper is to show that kck→0 is a necessary and sufficient condition for the uniform convergence of series ∑k=1∞‍cksin⁡kθ in θ∈[0,π]. However for ∑k=1∞‍cksin⁡k2θ it is not true in θ∈[0,π].

The purpose of this paper is to analyze the degree of approximation of a function \(\overline f\) that is a conjugate of a function \(f\) belonging to the Lipschitz class by Hausdorff means of a conjugate series of the Fourier series.

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Chaundy and Jolliffe proved that if a n is a nonnegative, nonincreasing real sequence, then series ∑ a n sin n x converges uniformly if and only if n a n ⟶ 0 . The purpose of this paper is to show that if n a n is nonincreasing and n a n ⟶ 0 , then the series f x = ∑ a n sin n x can be differentiated term-by-term on c , d for c , d > 0 . However, f ′ 0 may not exist.