Flumioxazin and S-metolachlor are widely used in conventional sweetpotato production in North Carolina and other states; however, some growers have recently expressed concerns about potential effects of these herbicides on sweetpotato yield and quality. Previous research indicates that activated charcoal has the potential to reduce herbicide injury. Field studies were conducted in 2021 and 2022 to determine whether flumioxazin applied preplant and S-metolachlor applied before and after transplanting negatively affect sweetpotato yield and quality when activated charcoal is applied with transplant water. The studies evaluated five herbicide treatments and two activated charcoal treatments. Herbicide treatments included two flumioxazin rates, one S-metolachlor rate applied immediately before and immediately after transplanting, and no herbicide. Charcoal treatments consisted of activated charcoal applied at 9 kg ha−1, and no charcoal. No visual injury from herbicides or charcoal was observed. Likewise, no effect of herbicide or charcoal treatment on no. 1, marketable (sum of no. 1 and jumbo grades), or total yield (sum of canner, no. 1, and jumbo grades) was observed. Additionally, shape analysis conducted on calculated length-to-width ratio (LWR) for no. 1 sweetpotato roots found no effect from flumioxazin at either rate on sweetpotato root shape. However, both S-metolachlor treatments resulted in lower LWR of no. 1 sweetpotato roots in 2021. Results are consistent with prior research and indicate that flumioxazin and S-metolachlor are safe for continued use on sweetpotato at registered rates.

Greenhouse trials were conducted to determine the response of stevia to reduced-risk synthetic and nonsynthetic herbicides applied over-the-top post-transplant. In addition, field trials were conducted with stevia grown in a polyethylene mulch production system to determine crop response and weed control in planting holes to reduced-risk synthetic and nonsynthetic herbicides applied post-transplant directed. Treatments included caprylic acid plus capric acid, clove oil plus cinnamon oil, d-limonene, acetic acid (200 grain), citric acid, pelargonic acid, eugenol, ammonium nonanoate, and ammoniated soap of fatty acids. Stevia yield (dry aboveground biomass) in the greenhouse was reduced by all herbicide treatments. Citric acid and clove oil plus cinnamon oil were the least injurious, reducing yield by 16% to 20%, respectively. In field studies, d-limonene, pelargonic acid, ammonium nonanoate, and ammoniated soap of fatty acids controlled Palmer amaranth (>90% 1 wk after treatment (WAT). In field studies caprylic acid plus capric acid, pelargonic acid, and ammonium nonanoate caused >30% injury to stevia plants at 2 WAT, and d-limonene, citric acid, acetic acid, and ammoniated soap of fatty acids caused 18% to 25% injury 2 WAT. Clove oil plus cinnamon oil and eugenol caused <10% injury. Despite being injurious, herbicides applied in the field did not reduce yield compared to the nontreated check. Based upon yield data, these herbicides have potential for use in stevia; however, these products could delay harvest if applied to established stevia. In particular, clove oil plus cinnamon oil has potential for use for early-season weed management for organic production systems. The application of clove oil plus cinnamon oil over-the-top resulted in <10% injury 28 d after treatment (DAT) in the greenhouse and 3% injury 6 WAT postemergence-directed in the field. In addition, this treatment provided 95% control of Palmer amaranth 4 WAT.

Field studies were conducted to assess the efficacy of physical weed management of Palmer amaranth management in cucumber, peanut, and sweetpotato. Treatments were arranged in a 3 × 4 factorial in which the first factor included a treatment method of electrical, mechanical, or hand-roguing Palmer amaranth control and the second factor consisted of treatments applied when Palmer amaranth was approximately 0.3, 0.6, 0.9, or 1.2 m above the crop canopy. Four wk after treatment (WAT), the electrical applications controlled Palmer amaranth at least 27 percentage points more than the mechanical applications when applied at the 0.3- and 0.6-m timings. At the 0.9- and 1.2-m application timings 4 WAT, electrical and mechanical applications controlled Palmer amaranth by at most 87%. Though hand removal generally resulted in the greatest peanut pod count and total sweetpotato yield, mechanical and electrical control resulted in similar yield to the hand-rogued plots, depending on the treatment timing. With additional research to provide insight into the optimal applications, there is potential for electrical control and mechanical control to be used as alternatives to hand removal. Additional studies were conducted to determine the effects of electrical treatments on Palmer amaranth seed production and viability. Treatments consisted of electricity applied to Palmer amaranth at first visible inflorescence, 2 wk after first visible inflorescence (WAI) or 4 WAI. Treatments at varying reproductive maturities did not reduce the seed production immediately after treatment. However, after treatment, plants primarily died and ceased maturation, reducing seed production assessed at 4 WAI by 93% and 70% when treated at 0 and 2 WAI, respectively. Treatments did not have a negative effect on germination or seedling length.

The strong maximum principle for harmonic functions is usually arrived at by appealing to the mean value theorem (c.f. [2], p. 53). It is also of course possible simply to appeal to the Hopf maximum principle [2], but using sledge hammers to kill flies is generally viewed as aesthetically unpleasing. In contrast to the case of harmonic functions, the only proof of the strong maximum principle for the heat equation that is known to me is to invoke Nirenberg's strong maximum principle for parabolic equations [2]. As in the case of harmonic functions, it seems desirable to provide a direct proof of this result without having to go through the subtle comparison arguments that are employed in the more general case. The purpose of this note is to provide a proof of the strong maximum principle for the heat equation based on a mean value theorem for solutions of the heat equation which we derive below. Such an approach provides a straightforward and simple proof of the strong maximum principle which avoids most of the detailed estimates of the proof of the maximum principle for more general parabolic equations. Unfortunately the proof of the maximum principle for the heat equation using the mean value theorem is not as short as the proof in the corresponding case of harmonic functions. It nevertheless seems worthwhile to show that such an alternate proof is possible, and it is to this purpose that we address this paper.

The inverse problem we will consider in this paper has its origins in the following problem connected with the scattering of acoustic waves in a nonhomogeneous medium. Let an incoming plane acoustic wave of frequency ω moving in the direction of the z axis be scattered off a “soft” sphere Ω of radius one which is surrounded by a pocket of rarefied or condensed air in which the local speed of sound is given by c(r) where r = |x| for x ∈ R3. Let us(x)eiωt be the velocity potential of the scattered wave and let r, θ, φ be spherical coordinates in R3. Then from a knowledge of the far field pattern f(θ, φ, λ) for λ = ω/c0 contained in some finite interval 0 < λ0 ≤ λ ≤ λ1,, we would like to determine the unknown function c(r).

In this note we shall consider the problem of uniquely continuing solutions of the parabolic equation

across an analytic arc σ: x=s1(t) satisfies the boundary data

We assume that u(x,t) is a classical solution of (1) in the domain D ={(x,t): s1(t)< x < s 2(t), 0 < t < t0}, continuously differentiate in D ∪ σ and define the “reflection” of D across σ by

In this paper, we shall obtain two results on the class of far field patterns corresponding to the scattering of time harmonic acoustic plane waves by an inhomogeneous medium of compact support. Although the problem of characterizing the class of far field patterns is of basic importance in inverse scattering theory, very little is known about this class other than the fact that the far field patterns are entire functions of their independent (complex) variables for each positive fixed value of the wave number. In particular, the class of far field patterns is not all of L2(∂Ω) where ∂Ω is the unit sphere and this implies that the inverse scattering problem is improperly posed since the far field patterns are, in practice, determined from inexact measurements. The purpose of this paper is to show that while the class of far field patterns corresponding to the scattering of time harmonic plane waves by an inhomogeneous medium is not all of L2(∂Ω), it is dense in L2(∂Ω) for sufficiently small values of the wave number. In addition, a related result will be obtained for a special translation of the class of far field patterns. Analogous results for the scattering of time harmonic acoustic waves by a homogeneous scattering obstacle have recently been obtained by Colton [1], Colton and Kirsch [2], Colton and Monk [3, 4] and Kirsch [8].

A key step in establishing the validity of the linear sampling method of determining an unknown scattering obstacle $D$ from a knowledge of its far-field pattern is to prove that solutions of the Helmholtz equation in $D$ can be approximated in $H^1(D)$ by Herglotz wave functions.

To this end we establish the important property that the set of Herglotz wave functions is dense in the space of solutions of the Helmholtz equation with respect to the Sobolev space $H^1(D)$ norm.

AMS 2000 Mathematics subject classification: Primary 35R30. Secondary 35P25