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Frank Bonsall played a significant role in the mathematical life of the United Kingdom in the decades following the Second World War. He had a particular impact in Scotland and the north of England, especially in research and graduate education. His research interests focused primarily on functional analysis, the area of mathematics that brings together various strands of analysis under a single abstract framework, and on the related theory of linear operators on Banach spaces. He influenced a generation of young mathematicians with the elegance of his written and oral expositions, both of his own research and that of others. The quality of his caring and thorough research supervision was reflected in his many PhD students who would continue in research and go on to successful academic careers in their own right, both in the United Kingdom and beyond.

Patients treated for breast cancer are at risk of developing breast cancer-related lymphedema (BCRL). A significant proportion of patients treated for breast cancer are opting to undergo a contralateral prophylactic mastectomy (CPM). Currently, it remains unclear as to whether the relative volume change (RVC) equation may be used as an alternative to the weight adjusted change (WAC) equation to quantify BCRL in patients who undergo CPM. In order to simplify BCRL screening, our cohort of patients who underwent a CPM (n=310) was matched by BMI to a subset of patients who underwent unilateral breast surgery (n=310). Arm volume measurements were obtained via an optoelectronic perometer preoperatively, postoperatively, and in the follow-up setting every 6-12 months. The correlation of ipsilateral RVC and WAC values for those who underwent bilateral surgery was calculated (r=0.60). Contralateral WAC values for patients in both cohorts were compared, and there was no significant difference between the two distributions in variance (p=0.446). The RVC equation shows potential to be used to quantify ipsilateral postoperative arm volume changes for patients who undergo a CPM. However, a larger trial in which RVC and WAC values are prospectively assessed is needed.

Breast cancer-related lymphedema (BCRL) affects more than one in five women treated for breast cancer, and women remain at lifelong risk. Screening for BCRL is recommended by several national and international organizations for women at risk of BCRL, and multiple methods of objective screening measurement exist. The goal of this study was to compare the use of perometry and bioimpedance spectroscopy (BIS) for early identification of BCRL in a cohort of 138 prospectivelyscreened patients. At each screening visit, a patient's relative volume change (RVC) from perometer measurements and change in L-Dex from baseline (ΔL-Dex) using BIS was calculated. There was a negligible correlation between RVC and ΔL-Dex (r=0.195). Multiple thresholds of BCRL were examined: RVC ≥5% and ≥10% as well as and ΔL-Dex ≥6.5 and ≥10. While some patients developed an elevated RVC and ΔL-Dex, many demonstrated elevations in only one threshold category. Moreover, the majority of patients with RVC ≥5%, ΔL-Dex ≥6.5, or ΔL-Dex ≥10 regressed to non-elevated measurements without intervention. These findings suggest a role for combining multiple screening methods for early identification of BCRL; furthermore, BCRL diagnosis must incorporate patient symptoms and clinical evaluation with objective measurements obtained from techniques such as perometry and bioimpedance spectroscopy.

Frank Bonsall played a significant role in the mathematical life of the United Kingdom in the decades following the Second World War. He had a particular impact in Scotland and the north of England, especially in research and graduate education. His research interests focused primarily on functional analysis, the area of mathematics that brings together various strands of analysis under a single abstract framework, and on the related theory of linear operators on Banach spaces. He influenced a generation of young mathematicians with the elegance of his written and oral expositions, both of his own research and that of others. The quality of his caring and thorough research supervision was reflected in his many PhD students who would continue in research and go on to successful academic careers in their own right, both in the United Kingdom and beyond.

It is shown that for $11$ it is even shown that $\|(\sum_{k=1}^n|R_k(f,g)|^2)^{1/2}\|_{L^{p_3}(\R^n)}\le C_2 \|f\|_{L^{p_1}(\R^n)}\|g\|_{L^{p_2}(\R^n)}$ for some constant $C_2$ independent of the dimension.

The disposition and metabolism of prasugrel, a thienopyridine prodrug and a potent inhibitor of platelet aggregation in vivo, were investigated in mice, rats, and dogs. Prasugrel was rapidly absorbed and extensively metabolized. In the mouse and dog, maximum plasma concentration of radioactivity was observed in less than 1 h after an oral [14C]prasugrel dose. Most of the administered prasugrel dose was recovered in the faeces of rats and dogs (72% and 52-73%, respectively), and in mice urine (54%). Prasugrel is hydrolysed by esterases to a thiolactone, which is subsequently metabolized to thiol-containing metabolites. The main circulating thiol-containing metabolite in the three animal species is the pharmacologically active metabolite, R-138727. The thiol-containing metabolites are further metabolized by S-methylation and conjugation with cysteine.

Leaf wetness duration (LWD) is related to plant disease occurrence and is therefore a key parameter in agrometeorology. As LWD is seldom measured at standard weather stations, it must be estimated in order to ensure the effectiveness of warning systems and the scheduling of chemical disease control. Among the models used to estimate LWD, those that use physical principles of dew formation and dew and/or rain evaporation have shown good portability and sufficiently accurate results for operational use. However, the requirement of net radiation (Rn) is a disadvantage foroperational physical models, since this variable is usually not measured over crops or even at standard weather stations. With the objective of proposing a solution for this problem, this study has evaluated the ability of four models to estimate hourly Rn and their impact on LWD estimates using a Penman-Monteith approach. A field experiment was carried out in Elora, Ontario, Canada, with measurements of LWD, Rn and other meteorological variables over mowed turfgrass for a 58 day period during the growing season of 2003. Four models for estimating hourly Rn based on different combinations of incoming solar radiation (Rg), airtemperature (T), relative humidity (RH), cloud cover (CC) and cloud height (CH), were evaluated. Measured and estimated hourly Rn values were applied in a Penman-Monteith model to estimate LWD. Correlating measured and estimated Rn, we observed that all models performed well in terms of estimating hourly Rn. However, when cloud data were used the models overestimated positive Rn and underestimated negative Rn. When only Rg and T were used to estimate hourly Rn, the model underestimated positive Rn and no tendency was observed for negative Rn. The best performance was obtained with Model I, which presented, in general, the smallest mean absolute error (MAE) and the highest C-index. When measured LWD was compared to the Penman-Monteith LWD, calculated with measured and estimated Rn, few differences were observed. Both precision and accuracy were high, with the slopes of the relationships ranging from 0.96 to 1.02 and R2 from 0.85 to 0.92, resulting in C-indices between 0.87 and 0.93. The LWD mean absolute errors associated with Rn estimates were between 1.0 and 1.5 h, which is sufficient for use in plant disease management schemes.

It is shown that an invertible isometry on lp, where \(1\leq p< \infty \;\mathrm{and}\;p\neq2\), is a scalar-type spectral operator provided its spectrum is a proper subset of the unit circle. A similar, though weaker, analysis is also considered for invertible isometries on more general Lp spaces. These results are used to give several examples of invertible operators U on Lp spaces, where \(p\;\in\;(1,\infty)\;\mathrm{and}\;p \leq\;2\), such that \(\mathrm{sup}_{n\in \mathbb{Z}}\|U^{n}\| \; < \infty\) but U is not similar to an invertible isometry. This contrasts with the situation on Hilbert space, where the condition \(\mathrm{sup}_{n\in \mathbb{Z}}\|U^{n}\| \; < \infty\) on an invertible operator U implies that U is similar to a unitary operator.

We develop a general condition for automatically discretizing strong type bisublinear maximal estimates that arise in the context of the real line. In particular, this method applies directly to Michael Lacey’s strong type boundedness results for the bisublinear maximal Hilbert transform and for the bisublinear Hardy-Littlewood maximal operator, furnishing the counterpart of each of these two results (without changes to the range of exponents) for the sequence spaces $\ell^p ({\Bbb Z}).$ We then take up some transference applications of discretized maximal bisublinear operators to maximal estimates and almost everywhere convergence in Lebesgue spaces of abstract measures. We also broaden the scope of such applications, which are based on transference from ${\Bbb Z},$ by developing general methods for transplanting bisublinear maximal estimates from arbitrary locally compact abelian groups.

In this article we obtain the boundedness of the periodic, discrete and ergodic bilinear Hilbert transform, from $L^{p_1}\times L^{p_2}$ into $L^{p_3}$ , where $1/p_1+ 1/p_2=1/p_3$, $p_1, p_2 > 1$, and $p_3\ge 1$ . The main techniques are a bilinear version of the transference method of Coifman and Weiss and certain discretization of bilinear operators. In the periodic case, we also obtain the boundedness for $2/3

Given a compact connected abelian group G, its dual group Γ can be ordered (in a non-canonical way) so that it becomes an ordered group. It is known that, for any such ordering on Γ and p in the range 1

For 1 q 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the directionq > 1. More- over, when taken in conjunction with vector-valued transference, this Mq(T)-multiplier result shows that if X2I, and U is an invertible power-bounded operator on X, then U has an Mq(T)-functional calculus for an appropriate range of values of q > 1. The class I includes, in particular, all closed subspaces of the von Neumann{Schatten p-classesCp (1 < p

In this paper we establish dimension freeL p (ℝ n ,|x|α) norm inequalities (1

Let T = {Ts}s∈R be a strongly continuous one-parameter group of positive operators acting on L(μ), where μ is a σ-finite measure and 1 ≤ p < ∞. We develop a weighted ergodic theory associated with T which parallels the classical theory of Ap-weights. A weighted transference principle is obtained in this setting and used to establish dimension free weighted inequalities for certain classical operators, including the higher order Riesz transforms.

Suppose that (Ω,μ) is a σ-finite measure space, and 1 < p < ∞. Let T:Lp(μ → L p(μ) be a bounded invertible linear operator such that T and T−1 are positive. Denote by \({\mathfrak{E}}\)n(T) the nth two-sided ergodic average of T, taken in the form of the nth (C,1) mean of the sequence {Tj+T−j}j =1∞. Martin-Reyes and de la Torre have shown that the existence of a maximal ergodic estimate for T is characterized by either of the following two conditions: (a) the strong convergence of En(T)n=1∞; (b) a uniform App estimate in terms of discrete weights generated by the pointwise action on Ω of certain measurable functions canonically associated with T. We show that strong convergence of the (C,2) means of {Tj+T−j}j=1∞ already implies (b). For this purpose the (C,2) means are used to set up an `averaged' variant of the requisite uniform Ap weight estimates in (b). This result, which can be viewed as a Tauberian-Type replacement of (C,1) means by (C,2) means in (a), leads to a spectral-theoretic characterization of the maximal ergodic estimate by application of a recent result of the authors establishing the strong convergence of the (C,2)-weighted ergodic means for all trigonometrically well-bounded operators. This application also serves to equate uniform boundedness of the rotated Hilbert averages of T with the uniform boundedness of the ergodic averages En(T)n = 1∞.

It is well-known that the fundamental solution of $$ u_t(n,t)= u(n+1,t)-2u(n,t)+u(n-1,t), \quad n\in\mathbb{Z}, $$ with $u(n,0) =\delta_{nm}$ for every fixed $m \in\mathbb{Z}$, is given by $u(n,t) = e^{-2t}I_{n-m}(2t)$, where $I_k(t)$ is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series $$ W_tf(n) = \sum_{m\in\mathbb{Z}} e^{-2t} I_{n-m}(2t) f(m). $$ By using semigroup theory, this formula allows us to analyze some operators associated with the discrete Laplacian. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted $\ell^p(\mathbb{Z})$-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. Interestingly, it is shown that the Riesz transforms coincide essentially with the so called discrete Hilbert transform defined by D. Hilbert at the beginning of the XX century. We also see that these Riesz transforms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel functions., Comment: 21 pages. To appear in Journal d'Analyse Mathematique

An example is given of an operator weight W that satisfies the dyadic operator Hunt–Muckenhoupt–Wheeden condition [Aopf ]d2 for which there exists a dyadic martingale transform on L2 (W) that is unbounded. The construction relates weighted boundedness to the boundedness of dyadic vector Hankel operators.

Let U be a trigonometrically well-bounded operator on a Banach space X, and denote byfAn(U)g 1=1 the sequence of (C; 2) weighted discrete ergodic averages of U, that is, An(U) = 1 n X 0

AC-operators are a generalization in the context of well-boundedness of normal operators on Hilbert space. It was shown by Doust and Walden that compact AC-operators have a representation as a conditionally convergent sum reminiscent of the spectral representations for compact normal operators. In this representation, the eigenvalues must be taken in a particular order to ensure convergence of the sum. Here we show that one cannot replace the ordering given by Doust and Walden by the more natural one suggested in their paper.

We introduce and study the class of mean2-bounded operators on Hilbert space. These operators, which are characterized by a uniform boundedness condition on the ''quadratic averages'' of their powers, are intimately related to operator-valued weight sequences and weighted norm inequalities. Mean2-bounded operators exhibit a lively interplay with the discrete Hilbert transform in an array of settings. Under appropriate circumstances, mean2-bounded operators have a rich spectral theory leading to powerful transference properties.

Hourly meteorological observations over a potato crop field were used to validate a biophysical model and different thresholds of relative humidity (RH) to simulate the onset. cessation and total wetness duration (WD). The model showed the capability to simulate multiple wet and dry conditions as well as prolonged moist conditions with a mean absolute error of less than an hour. The deviation between measured and estimated onset and total WD was more pronounced when only RH was used. However, under the prevailing agroclimatic conditions of potato growing regions in India, 80% RH threshold may adequately be accurate to estimate WD for many weather-based disease management advisories.

Olanzapine metabolism was investigated using incubation of olanzapine with human liver slices. The intent of the investigation was to identify olanzapine metabolites and determine if the human liver slice incubations could potentially produce quantities of the olanzapine glucuronides for future studies. Along with known Phase 1 olanzapine metabolites, N-desmethyl-, 2-hydroxymethyl-, and 4'-N-oxide-, a new hydroxylated species was detected. Detection of Phase 2 metabolites included known N-10-glucuronides, a quaternary glucuronide and a novel glucuronide conjugate. This investigation showed the feasibility of using human liver slices to produce sufficient quantities of olanzapine glucuronides for further studies.

If 1 2 such that Mq(R) ⊆ Mp,ω(R) for 1 ≤ q

The onset and cessation of surface wetness on maize ears were simulated with six models, using hourly meteorological data, to examine the linkage between wetness duration and possible forecasting of fungal infections that produce mycotoxins. Two threshold models (using relative humidity and dew point temperature), one regression model (using humidity and wind speed), and three physical models based on the energy balance approach, were compared. Also, spatial and temporal variability in wetness duration was measured and simulated at three sites located at distances up to 29 km from a central weather station. The estimated wetness values were compared with observations from cylindrical wetness sensors placed near ear level in maize canopies. The results relate to potential mycotoxin warning systems and indicate that threshold models can provide reasonable estimates of ear wetness duration in this region, that a comprehensive physical model can give superior estimates, and that wetness estimates made from a central weather station data can be extended to nearby crop fields with a moderate degree of confidence. Key words: epidemiology, mycotoxins, surface wetness models

Let E and F be two commuting bounded Boolean algebras of projections on a Banach spaceX. Various geometric conditions onXare established which ensure that the Boolean algebra of projections generated by E and F is also bounded. In particular, this is the case ifXis a Banach lattice or a subspace of ap-concave Banach lattice, where 1⩽p

Suppose that ( Ω , M , μ ) (\Omega ,\mathcal {M},\mu ) is a σ \sigma -finite measure space, 1 > p > ∞ 1>p>\infty , and T : L p ( μ ) → L p ( μ ) T: L^{p}(\mu )\to L^{p}(\mu ) is a bounded, invertible, separation-preserving linear operator such that the linear modulus of T T is mean-bounded. We show that T T has a spectral representation formally resembling that for a unitary operator, but involving a family of projections in L p ( μ ) L^{p}(\mu ) which has weaker properties than those associated with a countably additive Borel spectral measure. This spectral decomposition for T T is shown to produce a strongly countably spectral measure on the “dyadic sigma-algebra” of T \mathbb {T} , and to furnish L p ( μ ) L^{p}(\mu ) with abstract analogues of the classical Littlewood-Paley and Vector-Valued M. Riesz Theorems for ℓ p ( Z ) \ell ^{p}(\mathbb {Z}) .

This paper gives a survey of some recent developments in the spectral theory of linear operators on Banach spaces in which the Hilbert transform and its abstract analogues play a fundamental role.

The influence of temperature and vapor pressure deficit (VPD) on the rate of predation by the predatory mite, Amblyseius cucumeris (Oudemans) on Frankliniella occidentalis (Pergande) was determined under controlled laboratory conditions. The survival of first instars of F. occidentalis was initially determined by themselves. Then the number of first instars that were killed by a single female adult of A. cucumeris was determined for the same density of thrips. The differences in the mortality between the two experiments were used to calculate the rate of predation by A. cucumeris under a range of temperature and VPD regimes. Rate of predation was expressed as the number of thrips killed per h to account for the different time periods that the trials lasted. A quadratic model was fitted to the data. At a constant temperature, the rate of predation decreased with increasing VPD for VPDs < 1.24 to 1.44 kPa. Above these VPDs, the rate of predation started to increase again. At a constant VPD, the rate of predation decreased slightly at the lower temperatures and increased at the higher temperatures. The rate of change was dependent upon the VPD. By using the quadratic model, the rate of predation can be predicted for the range of hygrothermal conditions that would be encountered in the greenhouse. The optimal condition for maximum rate of predation by A. cucumeris on first instars of F. occidentalis in the greenhouse is a VPD < 0.75 kPa at the recommended production temperatures and VPDs (17 to 25°C and < 1.5 kPa).

Let G be a locally compact abelian group with character group Γ. We study the interplay of boundedness properties for suitably related maximal operators defined by (weak type or strong type) multipliers for G, its subgroups, and its Bohr compactification b(G). These considerations lead to weak type and strong type maximal estimates which generalize fundamental theorems of de Leeuw and Saeki concerning strong type norms of single multipliers. Suppose that 1 ≤ p < ∞, and M is the maximal operator on Lp(G) defined by a sequence {ψn}∞n = 1 of strong type Fourier multipliers which are continuous functions on γ. Our main result establishes that M is of weak type (p, p) on Lp(G) if and only if the corresponding maximal operator M# on Lp(b(G)) is of weak type (p, p). This provides a counterpart for locally compact abelian groups of E. M. Stein′s Continuity Principle for compact groups, since the latter characterizes the weak (p, p) boundedness of M# when 1 ≤ p ≤ 2.

A simple, relatively inexpensive, and effective method for the control of relative humidity in an environmental chamber was developed and evaluated. Relative humidity was maintained at 90% ± 0.4% and 95.2% ± 0.5% through the use of a micrologger-controlled ultrasonic mister that cycled between relatively narrow set points. To maintain consistent relative humidity, misters were activated at approximately 5- and 3-min intervals at 90% and 95% relative humidity, respectively. Air temperatures of 20–28°C were maintained at ±0.5°C by regulation of growth room controls. This environmental chamber has considerable utility for investigations of the influence of relative humidity and water potential on plants and plant-associated microorganisms.

We describe sufficient conditions for transferring from locally compact abelian groups to measure spaces the weak-type bounds of maximal operators defined by multipliers of weak type. This leads to homomorphism theorems for maximal multiplier operators.

In the context of a locally compact abelian group, we establish maximal theorem counterparts for weak type (1,1) multipliers of the classical de Leeuw theorems for individual strong multipliers. Special methods are developed to handle the weak type (1,1) estimates involved since standard linearization methods such as Lorentz space duality do not apply to this case. In particular, our central result is a maximal theorem for convolutions with weak type (1,1) multipliers which opens avenues of approximation. These results complete a recent series of papers by the authors which extend the de Leeuw theorems to a full range of strong type and weak type maximal multiplier estimates in the abstract setting.

Let Δ(x) = max {1 - ¦x¦, 0} for all x ∈ ℝ, and let ξ[0,1) be the characteristic function of the interval 0 ≤x < 1. Two seminal theorems of M. Jodeit assert that A and ξ[0,1) act as summability kernels convertingp-multipliers for Fourier series to multipliers forL P (ℝ). The summability process corresponding to Δ extendsL P (T)-multipliers from ℤ to ℝ by linearity over the intervals [n, n + 1],n ∈ ℤ, when 1 ≤p < ∞, while the summability process corresponding to ξ[0,1) extends LP(T)-multipliers by constancy on the intervals [n, n + 1),n ∈ ℤ, when 1

Controlled environment studies have shown that peanut (Arachis hypogaea L.) leaf CO2 exchange rates (CER) are sensitive to night temperatures OAC Garroy > all other genotypes. The cultivar OAC Ruby showed a reduction in CER of approximately 25% when Tmin fell from 20 to 5 degrees C. In contrast, Early Bunch, Chico, VA910212, and OAC Tango did not differ from each other and were relatively sensitive to low Tmin, with CER reduced by approximately 55% for the same reduction in Tmin. After Tmin of 20 degrees C, however, CER of sensitive genotypes was 17% higher than that of OAC Ruby. Canopy minimum temperatures at Delhi were 1 to 3 degrees C cooler than Tmin, primarily due to radiative cooling. Microclimate differences between environments will affect the magnitude of differences between Tmin and Tmin (leaf) and therefore affect the utility of Tmin-CER relationships developed here.

The deposition of ozone onto hypostomatous red maple and amphistomatous hybrid poplar leaves was studied under controlled environments. Under similar environmental conditions, the ozone deposition was substantially greater onto poplar than onto maple leaves. Two kinds of surface wetness (simulating dew and raindrops) were artificially created on leaves to determine how deposition was affected. The ozone deposition onto wet maple leaves markedly increased almost immediately after water spraying. The increases were measured during both daytime and nighttime conditions. Nighttime measurements revealed that the enhanced ozone deposition onto wet maple leaves was largely controlled by the chemistry of the solution which eventually developed while leaves remained wet. Unlike maple leaves, small ozone deposition reductions were measured after poplar leaves became wet during daytime conditions. In addition, little ozone deposition was detected onto wet poplar leaves during the night, indicating that leaf water chemistry was less important for poplar than for maple leaves.

The influence of temperature and vapor pressure deficit (VPD) were determined on larval, pupal, and adult stages of nonfeeding Frankliniella occidentalis (Pergande) under controlled laboratory conditions and on first instars confined to cucumber leaves under controlled greenhouse conditions. The influence of temperature and VPD on sanitation clean-up for control of thrips at the end of the growing season in the greenhouse was also determined. In laboratory trials, the larval stages exhibited the lowest percentages of survival under the range of temperatures and VPDs tested. The adult stage had >80% survival except with high VPDs at 25, 30, and 35°C. Pupae had 100% survival at all temperatures except 35°C, at which survival decreased to ≍60%at the higher VPDs. A quadratic model was fitted to the response surfaces of the larval and adult stages. By using the quadratic model, it is possible to predict the response of the different stages of F. occidentalis to a range of hygrothermal conditions. Cage trials of first instars on cucumber leaves in the greenhouse also showed that high temperature and VPD will decrease the percentage survival of F. occidentalis . Sanitation trials demonstrated that 40°C and VPD ≥ 4.76 kPa will control F. occidentalis at the end of the crop season when the greenhouse is being prepared for the next crop.

Let X be a closed subspace of LP(μ), where μ is an arbitrary measure and 1 < p < ∞. By extending the scope of spectral integration, we show that every invertible power-bounded linear mapping of X into X has a functional calculus implemented by the algebra of complex-valued functions on the unit circle satisfying the hypotheses of the Strong Marcinkiewicz Multiplier Theorem. This result expands the framework of the Strong Marcinkiewicz Multiplier Theorem to the setting of abstract measure spaces.