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Passive acoustic mapping (PAM) is an algorithm that reconstructs the location of acoustic sources using an array of receivers. This technique can monitor therapeutic ultrasound procedures to confirm the spatial distribution and amount of microbubble activity induced. Current PAM algorithms have an excellent lateral resolution but have a poor axial resolution, making it difficult to distinguish acoustic sources within the ultrasound beams. With recent studies demonstrating that short-length and low-pressure pulses—acoustic wavelets—have the therapeutic function, we hypothesized that the axial resolution could be improved with a quasi-pulse-echo approach and that the resolution improvement would depend on the wavelet’s pulse length. This article describes an algorithm that resolves acoustic sources axially using time of flight and laterally using delay-and-sum beamforming, which we named axial temporal position PAM (ATP-PAM). The algorithm accommodates a rapid short pulse (RaSP) sequence that can safely deliver drugs across the blood–brain barrier. We developed our algorithm with simulations (k-wave) and in vitro experiments for one-, two-, and five-cycle pulses, comparing our resolution against that of two current PAM algorithms. We then tested ATP-PAM in vivo and evaluated whether the reconstructed acoustic sources mapped to drug delivery within the brain. In simulations and in vitro , ATP-PAM had an improved resolution for all pulse lengths tested. In vivo , experiments in mice indicated that ATP-PAM could be used to target and monitor drug delivery into the brain. With acoustic wavelets and time of flight, ATP-PAM can locate acoustic sources with a vastly improved spatial resolution.

Additive manufacturing is one of the most promising emerging technologies for building prosthetic sockets. However, there is no reliable way to estimate the factor of safety and the lifetime of 3D printed prosthetic sockets. Here, we explore 360° 3D digital image correlation (DIC) and discover how this new tool can increase our understanding of prosthetic structural failures. We establish that this new technology can dramatically improve the prosthetic design cycle by identifying local strain concentrations and by highlighting limitations in current simulated models. Overall, 360° 3D DIC technology empowers prosthetic engineers to characterize the performance of new materials and create innovative designs that are both safe and affordable.

International audience; Multinucleon transfer channels, populated in the inverse kinematic reaction ^197Au+^130Te at E_lab = 1.07 GeV, were studied by means of a high-resolution kinematic coincidence set-up composed of the large acceptance magnetic spectrometer PRISMA coupled to a new detection system acting as second arm. The comparison of the experimental mass-mass correlation matrix with Monte Carlo simulations allowed us to investigate the role of neutron evaporation in the population of neutron-rich heavy nuclei around A = 200.

We studied multinucleon transfer reactions in the 197Au+130Te system at Elab=1.07 GeV by employing the PRISMA magnetic spectrometer coupled to a coincident detector. For each light fragment we constructed, in coincidence, the distribution in mass of the heavy partner of the reaction. With a Monte Carlo method, starting from the binary character of the reaction, we simulated the de-excitation process of the produced heavy fragments to be able to understand their final mass distribution. The total cross sections for pure neutron transfer channels have also been extracted and compared with calculations performed with the grazing code.

Background: Recent measurements of fusion cross sections for the 28 Si + 28 Si system revealed a rather unsystematic behavior; i.e., they drop faster near the barrier than at lower energies. This was tentatively attributed to the large oblate deformation of 28 Si because coupled-channels (CC) calculations largely underestimate the 28 Si + 28 Si cross sections at low energies, unless a weak imaginary potential is applied, probably simulating the deformation. 30 Si has no permanent deformation and its low-energy excitations are of a vibrational nature. Previous measurements of this system reached only 4 mb, which is not sufficient to obtain information on effects that should show up at lower energies. Purpose: The aim of the present experiment was twofold: (i) to clarify the underlying fusion dynamics by measuring the symmetric case 30 Si + 30 Si in an energy range from around the Coulomb barrier to deep sub-barrier energies, and (ii) to compare the results with the behavior of 28 Si + 28 Si involving two deformed nuclei. Methods: 30 Si beams from the XTU tandem accelerator of the Laboratori Nazionali di Legnaro of the Istituto Nazionale di Fisica Nucleare were used, bombarding thin metallic 30 Si targets (50 µ g / cm 2 ) enriched to 99 . 64% in mass 30. An electrostatic beam deflector allowed the detection of fusion evaporation residues (ERs) at very forward angles, and angular distributions of ERs were measured. Results: The excitation function of 30 Si + 30 Si was measured down to the level of a few microbarns. It has a regular shape, at variance with the unusual trend of 28 Si + 28 Si. The extracted logarithmic derivative does not reach the L CS limit at low energies, so that no maximum of the S factor shows up. CC calculations were performed including the low-lying 2 + and 3 - excitations. Conclusions: Using a Woods-Saxon potential the experimental cross sections at low energies are overpredicted, and this is a clear sign of hindrance, while the calculations performed with a M3Y + repulsion potential nicely fit the data at low energies, without the need of an imaginary potential. The comparison with the results for 28 Si + 28 Si strengthens the explanation of the oblate shape of 28 Si being the reason for the irregular behavior of that system.

International audience; Carbon burning is a critical phase for nucleosynthesis in massive stars. The conditions for igniting this burning stage, and the subsequent isotope composition of the resulting ashes, depend strongly on the reaction rate for C12+C12 fusion at very low energies. Results for the cross sections for this reaction are influenced by various backgrounds encountered in measurements at such energies. In this paper, we report on a new measurement of C12+C12 fusion cross sections where these backgrounds have been minimized. It is found that the astrophysical S factor exhibits a maximum around Ecm=3.5–4.0 MeV, which leads to a reduction of the previously predicted astrophysical reaction rate.

International audience; The 12C+12C fusion reaction is one of the most important for nuclear astrophysics since it determines the carbon ignition in stellar environments. Two experiments which make use of the gamma-particle coincidence technique to measure the 12C+12C S-factors at deep sub barrier energies are discussed. Results are presented showing a decrease of the S-factor below Ec.m. = 3 MeV.

This contribution reports on the results of measurements of near- and sub-barrier fusion cross sections in the system $^{30}$Si+$^{30}$Si performed at the Laboratori Nazionali di Legnaro of INFN. The $^{30}$Si beam of the XTU Tandem accelerator in the energy range of 47 - 90 MeV, was delivered on a metallic $^{30}$Si target (50 $\mu$g/cm$^{2}$) enriched to 99.64 $\%$ in mass 30. A beam electrostatic deflector was used for the detection of evaporation residues. The excitation function obtained for $^{30}$Si+$^{30}$Si has been compared with the previous data on $^{28}$Si+$^{28}$Si and Coupled Channels (CC) calculations using the M3Y+repulsion potential, taking into account the low lying 2$^+$ and 3$^-$ excitations. Reproducing the low-energy $^{28}$Si+$^{28}$Si excitation function was only possible by using a weak imaginary potential, probably simulating the oblate deformation of this nucleus. On the contrary, the data on $^{30}$Si+$^{30}$Si are nicely reproduced by the CC calculations without any imaginary potential ($^{30}$Si has a spherical shape). The astrophysical S-factor does not show a maximum, so that there is no evidence for hindrance in the measured energy range. The logarithmic derivative of the two excitation functions highlights the difference between the two systems. Even above the barrier the two systems behave differently. This is best seen by comparing the two barrier distributions where the high energy peak observed for $^{28}$Si+$^{28}$Si is not found for $^{30}$Si+$^{30}$Si. This is presumably due to the stronger couplings present in $^{28}$Si and further theoretical analyses are in progress.

Carbon burning plays a crucial role in stellar evolution, where this reaction is an important route for the production of heavier elements. A particle-γ coincidence technique that minimizes the backgrounds to which this reaction is subject and provides reliable cross sections has been used at the Argonne National Laboratory to measure fusion cross-sections at deep sub-barrier energies in the 12 C+12 C system. The corresponding excitation function has been extracted down to a cross section of about 6 nb. This indicates the existence of a broad S-factor maximum for this system. Experimental results are presented and discussed.

The near- and sub-barrier fusion excitation function has been measured for the system 30 Si+30 Si at the Laboratori Nazionali di Legnaro of INFN, using the 30 Si beam of the XTU Tandem accelerator in the energy range 47 - 90 MeV. A set-up based on a beam electrostatic deflector was used for detecting fusion evaporation residues. The measured cross sections have been compared to previous data on 28 Si+28 Si and Coupled Channels (CC) calculations have been performed using M3Y+repulsion and Woods-Saxon potentials, where the lowlying 2+ and 3− excitations have been included. A weak imaginary potential was found to be necessary to reproduce the low energy 28 Si+28 Si data. This probably simulates the effect of the oblate deformation of this nucleus. On the contrary, 30 Si is a spherical nucleus, 30 Si+30 Si is nicely fit by CC calculations and no imaginary potential is needed. For this system, no maximum shows up for the astrophysical S-factor so that we have no evidence for hindrance, as confirmed by the comparison with CC calculations. The logarithmic derivative of the two symmetric systems highlights their different low energy trend. A difference can also be noted in the two barrier distributions, where the high-energy peak present in 28 Si+28 Si is not observed for 30 Si+30 Si, probably due to the weaker couplings in last case.

International audience; We discuss nucleon transfer and fusion reactions of the $^{40}$ Ca+$^{58}$ Ni and $^{40}$ Ca+$^{64}$Ni systems with the coupled-channels approach. It is first shown that a simple treatment for the transfer in the coupled-channels method cannot reproduce simultaneously the transfer probabilities and the subbarrier enhancement of the fusion cross sections. An alternative scheme for the transfer, that takes into account other collective states in the pair transfer channel is proposed.

International audience; An experiment was performed at LNL to obtain information concerning the yield distributions in both the 132Sn (N = 82) and the 208Pb (N = 126) regions of the nuclide chart. The experimental set-up was composed of the large-acceptance magnetic spectrometer PRISMA to detect the light partner of the reaction, in conjunction with a recently installed ancillary set-up to detect the heavy partner.

Background: Heavy-ion fusion reactions at energies near the Coulomb barrier are influenced by couplings between the relative motion and nuclear intrinsic degrees of freedom of the colliding nuclei. The time-dependent Hartree-Fock (TDHF) theory, incorporating the couplings at the mean-field level, as well as the coupled-channels (CC) method are standard approaches to describe low energy nuclear reactions.Purpose: To investigate the effect of couplings to inelastic and transfer channels on the fusion cross sections for the reactions $^{40}\mathrm{Ca}+^{58}\mathrm{Ni}$ and $^{40}\mathrm{Ca}+^{64}\mathrm{Ni}$.Methods: Fusion cross sections around and below the Coulomb barrier have been obtained from CC calculations, using the bare nucleus-nucleus potential calculated with the frozen Hartree-Fock method and coupling parameters taken from known nuclear structure data. The fusion thresholds and neutron transfer probabilities have been calculated with the TDHF method.Results: For $^{40}\mathrm{Ca}+^{58}\mathrm{Ni}$, the TDHF fusion threshold is in agreement with the most probable barrier obtained in the CC calculations including the couplings to the low-lying octupole ${3}_{1}^{\ensuremath{-}}$ state for $^{40}\mathrm{Ca}$ and to the low-lying quadrupole ${2}_{1}^{+}$ state for $^{58}\mathrm{Ni}$. This indicates that the octupole and quadrupole states are the dominant excitations while neutron transfer is shown to be weak. For $^{40}\mathrm{Ca}+^{64}\mathrm{Ni}$, the TDHF barrier is lower than predicted by the CC calculations including the same inelastic couplings as those for $^{40}\mathrm{Ca}+^{58}\mathrm{Ni}$. TDHF calculations show large neutron transfer probabilities in $^{40}\mathrm{Ca}+^{64}\mathrm{Ni}$ which could result in a lowering of the fusion threshold.Conclusions: Inelastic channels play an important role in $^{40}\mathrm{Ca}+^{58}\mathrm{Ni}$ and $^{40}\mathrm{Ca}+^{64}\mathrm{Ni}$ reactions. The role of neutron transfer channels has been highlighted in $^{40}\mathrm{Ca}+^{64}\mathrm{Ni}$.

Fusion cross sections have been measured in the 40 Ca + 58 Ni and 40 Ca + 64 Ni systems at energies around and below the Coulomb barrier. The 40 Ca beam was delivered by the XTU Tandem accelerator of the Laboratori Nazionali di Legnaro and evaporation residues were measured at very forward angles with the LNL electrostatic beam deflector. Coupled-channels calculations were performed which highlight possible strong effects of neutron transfers on the fusion below the barrier in the 40 Ca + 64 Ni system. Microscopic time-dependent Hartree-Fock calculations have also been performed for both systems. Preliminary results are shown.

Background: No data on the fusion excitation function of $^{48}\mathrm{Ti}+^{58}\mathrm{Fe}$ in the energy region near the Coulomb barrier existed prior to the present work, while fusion of $^{58}\mathrm{Ni}+^{54}\mathrm{Fe}$ was investigated in detail some years ago, down to very low energies, and clear evidence of fusion hindrance was noticed at relatively high cross sections. $^{48}\mathrm{Ti}$ and $^{58}\mathrm{Fe}$ are soft and have a low-lying quadrupole excitation lying at $\ensuremath{\approx}800--900$ keV only. Instead, $^{58}\mathrm{Ni}$ and $^{54}\mathrm{Fe}$ have a closed shell (protons and neutrons, respectively) and are rather rigid.Purpose: We aim to investigate (1) the possible influence of the different structures of the involved nuclei on the fusion excitation functions far below the barrier and, in particular, (2) whether hindrance is observed in $^{48}\mathrm{Ti}+^{58}\mathrm{Fe}$, and to compare the results with current coupled-channels models.Methods: $^{48}\mathrm{Ti}$ beams from the XTU Tandem accelerator of INFN-Laboratori Nazionali di Legnaro were used. The experimental setup was based on an electrostatic beam separator, and fusion-evaporation residues (ERs) were detected at very forward angles. Angular distributions of ERs were measured.Results: Fusion cross sections of $^{48}\mathrm{Ti}+^{58}\mathrm{Fe}$ have been obtained in a range of nearly six orders of magnitude around the Coulomb barrier, down to $\ensuremath{\sigma}\ensuremath{\simeq}2 \ensuremath{\mu}\mathrm{b}$. The sub-barrier cross sections of $^{48}\mathrm{Ti}+^{58}\mathrm{Fe}$ are much larger than those of $^{58}\mathrm{Ni}+^{54}\mathrm{Fe}$. Significant differences are also observed in the logarithmic derivatives and astrophysical $S$ factors. No evidence of hindrance is observed, because coupled-channels calculations using a standard Woods-Saxon potential are able to reproduce the data in the whole measured energy range. Analogous calculations for $^{58}\mathrm{Ni}+^{54}\mathrm{Fe}$ predict clearly too large cross sections at low energies. The two fusion barrier distributions are wide and display a complex structure that is only qualitatively fit by calculations.Conclusions: It is pointed out that all these different trends originate from the dissimilar low-energy nuclear structures of the involved nuclei. In particular, the strong quadrupole excitations in $^{48}\mathrm{Ti}$ and $^{58}\mathrm{Fe}$ produce the relative cross section enhancement and make the barrier distribution $\ensuremath{\approx}2$ MeV wider, thus probably pushing the threshold for hindrance below the measured limit.

International audience; The cross section of the ^12C+^12C fusion reaction at low energies is of paramount importance for models of stellar nucleosynthesis in different astrophysical scenarios, such as Type Ia supernovae and Xray superbursts, where this reaction is a primary route for the production of heavier elements. In a series of experiments performed at Argonne National Laboratory, using Gammasphere and an array of Silicon detectors, measurements of the fusion cross section of ^12C+^12C were successfully carried out with the gamma and charged-particle coincidence technique in the center-of-mass energy range of 3-5 MeV. These were the first background-free fusion cross section measurements for ^12C+^12C at energies of astrophysical interest. Our results are consistent with previous measurements in the high-energy region; however, our lowest energy measurement indicates a fusion cross section slightly lower than those obtained with other techniques.

Measurements of fusion excitation functions of Ni58+Sn124 and Ni64+Sn124 are extended towards lower energy to cross sections of 1 μb and are compared to detailed coupled-channels calculations. The calculations clearly show the importance of including transfer reactions in a coupled-channels treatment for such heavy systems. This result is different from the conclusion made in a previous article which claimed that the influence of transfer on fusion is not important for fusion reactions of Ni+Sn. In the energy region studied in this experiment no indication of fusion hindrance has been observed, which is consistent with a systematic study of this behavior.

Measurements of fusion excitation functions of Ni-58 + Sn-124 and Ni-64 + Sn-124 are extended towards lower energy to cross sections of 1 mu b and are compared to detailed coupled-channels calculations. The calculations clearly show the importance of including transfer reactions in a coupled-channels treatment for such heavy systems. This result is different from the conclusion made in a previous article which claimed that the influence of transfer on fusion is not important for fusion reactions of Ni + Sn. In the energy region studied in this experiment no indication of fusion hindrance has been observed, which is consistent with a systematic study of this behavior.

The fusion excitation function of 48 Ti + 58 Fe has been measured in a wide energy range around the Coulomb barrier, covering 6 orders of magnitude of the cross sections. We present here the preliminary re- sults of this experiment, and a full comparison with the near-by system 58 Ni + 54 Fe where evidence of fusion hindrance shows up at relatively high cross sections. The sub-barrier cross sections of 48 Ti + 58 Fe are much larger than those of 58 Ni + 54 Fe. Significant differences are also observed in the logarithmic derivatives, astro- physical S-factors and fusion barrier distributions. The influence of low-energy nuclear structure on all these trends is pointed out and commented. Coupled-channels calculations using a Woods-Saxon potential are able to reproduce the experimental results for 48 Ti + 58 Fe. The logarithmic derivative of the excitation function is very nicely fit, and no evidence of hindrance is observed down to around 1 μb. The fusion barrier distribution is rather wide, flat and structureless. It is only in qualitative agreement with the calculated distribution.

International audience; Cluster resonances in light heavy-ion systems like $^{12}$C+$^{12}$C and $^{12}$C+$^{16}$O may have a major impact on astrophysics stellar scenarios. Resonant radiative capture reactions have been studied for these systems at energies at and slightly below their Coulomb barriers to investigate the possible $^{12}$C-$^{12}$Cand $^{12}$C-$^{16}$O molecular origin of the resonances. Spins have been attributed to the resonances and specificities of their γ-decay have been identified. At deep sub-barrier energies, a fusion cross section measurement using the particle-γ coincidence technique is discussed for the $^{12}$C+$^{12}$C system. A new project is presented to possibly extend the $^{12}$C+$^{12}$C low-energy S factor study.

International audience; Multineutron and multiproton transfer channels, populated in the inverse kinematics reaction 197Au+130Te at Elab=1.07 GeV, were measured at Laboratori Nazionali di Legnaro using the presently heaviest ion beam delivered by the PIAVE-ALPI accelerator complex and detecting both projectile-like and targetlike ions. To this end the large solid angle magnetic spectrometer PRISMA was coupled to a second arm for the detection of the heavy fragments in kinematic coincidence with the light ones selected and identified with the spectrometer. The data analysis is still in progress and will allow to compare the yields of both light and heavy partner with theoretical predictions performed with the GRAZING code to get quantitative information on transfer channels and the effect of evaporation and fission on the production rate of primary fragments. The mass integrated Z distribution, extracted from the experimental data, evidenced the population of proton pickup channels that, in conjunction with the neutron stripping ones from the 130Te, open the path for the production of neutron-rich heavy nuclei. In the following, we will present some preliminary results as well as details on the experimental configuration and perspectives for future investigations in the neutron-rich heavy region.

Background: The nuclear structure of colliding nuclei is known to influence the fusion process. Couplings of the relative motion to nuclear shape deformations and vibrations lead to an enhancement of the sub-barrier fusion cross section in comparison with the predictions of one-dimensional barrier penetration models. This enhancement is explained by coupled-channels calculations including these couplings. The sub-barrier fusion cross section is also affected by nucleon transfer channels between the colliding nuclei.Purpose: The aim of the present experiment is to investigate the influence of the projectile and target nuclear structures on the fusion cross sections in the $^{40}\mathrm{Ca}+^{58}\mathrm{Ni}$ and $^{40}\mathrm{Ca}+^{64}\mathrm{Ni}$ systems.Methods: The experimental and theoretical fusion excitation functions as well as the barrier distributions were compared for these two systems. Coupled-channels calculations were performed using the ccfull code.Results: Good agreement was found between the measured and calculated fusion cross sections for the $^{40}\mathrm{Ca}+^{58}\mathrm{Ni}$ system. The situation is different for the $^{40}\mathrm{Ca}+^{64}\mathrm{Ni}$ system where the coupled-channels calculations with no nucleon transfer clearly underestimate the fusion cross sections below the Coulomb barrier. The fusion excitation function was, however, well reproduced at low and high energies by including the coupling to the neutron pair-transfer channel in the calculations.Conclusions: The nuclear structure of the colliding nuclei influences the fusion cross sections below the Coulomb barrier for both $^{40}\mathrm{Ca}+^{58,64}\mathrm{Ni}$ systems. Moreover, we highlighted the effect of the neutron pair-transfer channel on the fusion cross sections in $^{40}\mathrm{Ca}+^{64}\mathrm{Ni}$.

Measurements of the excitation function for the fusion of $^{24}\mathrm{Mg}+^{30}\mathrm{Si}$ $(Q=17.89\text{ }\text{ }\mathrm{MeV})$have been extended toward lower energies with respect to previous experimental data. The $S$-factor maximum observed in this large, positive-$Q$-value system is the most pronounced among such systems studied thus far. The significance and the systematics of an $S$-factor maximum in systems with positive fusion $Q$ values are discussed. This result would strongly impact the extrapolated cross sections and reaction rates in the carbon and oxygen burnings and, thus, the study of the history of stellar evolution.

Measurements of the excitation function for the fusion of (24)Mg+(30)Si (Q=17.89 MeV)have been extended toward lower energies with respect to previous experimental data. The S-factor maximum observed in this large, positive-Q-value system is the most pronounced among such systems studied thus far. The significance and the systematics of an S-factor maximum in systems with positive fusion Q values are discussed. This result would strongly impact the extrapolated cross sections and reaction rates in the carbon and oxygen burnings and, thus, the study of the history of stellar evolution.

A general framework is presented for describing shortest curve algorithms and their time complexity in regions of the plane whose boundaries may be curved. An algorithm that accepts curved boundary Jordan regions along with given start and end points and produces the shortest curve between them is presented. Its time complexity is bounded by the product of the complexity of the region's boundary and that of the output shortest curve. (When the region is a simple polygon with N vertices, the time bound is O(Nk), where k is the number of vertices in the shortest curve.) A second algorithm produces shortest curves in multiply connected regions with possibly curved boundary.

We revisit the third fundamental theorem of Lie (Lie III) for finite dimensional Lie algebras in the context of infinite dimensional matrices., Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

Fusion cross sections have been measured in the 40 Ca + 58 Ni and 40 Ca + 64 Ni systems at beam energies ranging from Elab = 104.75 MeV to 153.5 MeV using the Laboratori Nazionali di Legnaro electrostatic deflector. Distributions of barriers have been extracted from the experimental data. Preliminary coupled channel calculations were performed and hints of effects of neutron transfers on the fusion below the barrier in the 40 Ca + 64 Ni are discussed.

In this note we present a natural generalization of a well-known result of Dunford and Pettis. At the original writing our results prompted us to conjecture a still stronger theorem which we communicated to, and which has subsequently been settled by, S. Fitzpatrick. Continued justification for this note stems from the simplicity of both statements and proofs contained herewith. Recall that a bounded subset D of a Banach space is said to be s-dentable if for each e > 0 there corresponds a point x, in D which cannot be written as a (possibly infinite) convex combination of points in D each of distance at least e from x,; D is subset s-dentable if each of its subsets is s-dentable. The notational convention employed throughout is that A denotes a weak* compact and norm separable subset of a dual Banach space Y*. Several corollaries, including that mentioned in the title, follow from Theorem 1.

The general framework of a Markov chain in a random environment is presented and the problem of determining extinction probabilities is discussed. An efficient method for determining absorption probabilities and criteria for certain absorption are presented in the case that the environmental process is a two-state Markov chain. These results are then applied to birth and death, queueing and branching chains in random environments.

It is well known that a compact convex subset C of a locally convex topological vector space is a simplex if and only if each point x of C admits a unique probability measure on the extreme points of C with barycenter x . An exact analog of this result is proved for a closed and bounded separable convex subset of a Banach space with the Radon-Nikodým Property, and a weaker analog is proved in the nonseparable case.

Natural questions are settled concerning shortest paths between given points of a Jordan region.

Each norm closed and bounded convex subset K of a separable dual Banach space is, according to a theorem of Bessaga and Pelczynski, the norm closed convex hull of its extreme points. It is natural to expect that this theorem may be reformulated as an integral representation theorem, and in this connection we have examined the extent to which the Choquet theory applies to such sets. Two integral representation theorems are proved and an example is included which shows that a sharp result obtains for certain noncompact sets. In addition, the set of extreme points of K is shown to be a-measurable for each finite regular Borel measure , hence eliminating certain possible measure-theoretic difficulties in proving a general integral representation theorem. The last section is devoted to the study of a class of extreme points (called pinnacle points) which share important geometric properties with extreme points of compact convex sets in locally convex spaces. A uniqueness result is included for certain simplexes all of whose extreme points are pinnacle points. Introduction. In 1966 Bessaga and Pelczynski [1] proved that each norm closed and bounded convex subset of a separable dual Banach space is the norm closed convex hull of its extreme points, thus providing a "Krein-Milman" type theorem for a certain collection of noncompact sets. (A more elementary proof of this result has been given in [11].) For compact convex sets (in locally convex spaces) the Krein-Milman theorem can easily be reformulated as an integral representation theorem, using measures which are supported by the closure of the set of extreme points (see for example [13, p. 6]). This formulation has a large number of applications to analysis, probability, etcetera. A much more precise kind of representation theorem is provided by Choquet theory, where the measures (at least in the metrizable case) are supported by the set of extreme points, not just the closure. (See [5] and [2]. For a comprehensive and readable account see [13].) The BessagaPelczynski theorem would probably find greater applicability if it too could be reformulated as an integral representation theorem. As a step towards solving this problem we investigate to what extent the Choquet theory applies to the sets described in the Bessaga-Pelczynski theorem. We take this opportunity to thank Professor R. R. Phelps for directing the author's thesis from which this paper is Received by the editors October 17, 1969. AMS 1970 subject classifications. Primary 46B99; Secondary 46G99, 28A40.