290 research outputs found
Nikodym boundedness property for webs in sigma-algebras
[EN] A subset B of an algebra A of subsets of Omega is said to have property N if a
B-pointwise bounded subset M of ba(A ) is uniformly bounded on A , where ba(A ) is
the Banach space of the real (or complex) finitely additive measures of bounded variation
defined on A with the norm variation. Moreover B is said to have property sN if for each
increasing countable covering (B_m)_m of B there exists B_n which has property N and B
is said to have property wN if given the increasing countable coverings (B_m_1 )_m_1 of B and
(B_m_1m_2...m_pm_(p+1) )_m_(p+1) of B_m_1m_2...m_p , for each p,m_i ∈ N, 1<= i <= p + 1, there exists
a sequence (n_i )_i such that each B_n_1n_2...n_r , r ∈ N, has property N. For a σ-algebra S of
subsets of Omega it has been proved that S has property N (Nikodym Grothendieck), property
sN (Valdivia) and property w(sN) (Kakol López-Pellicer). We give a proof of property
wN for a σ-algebra S which is independent of properties N and sN. This result and the
equivalence of properties wN and w2N enable us to give some applications to localization
of bounded additive vector measures.This work was supported for the second named author by the Spanish Ministerio de Economía y Competitividad under Grant MTM2014-58159-PLópez Alfonso, S.; Mas Marí, J.; Moll López, SE. (2016). Nikodym boundedness property for webs in sigma-algebras. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 110(2):711-722. https://doi.org/10.1007/s13398-015-0260-4S7117221102Diestel, J.: Sequences and Series in Banach Spaces. Springer, New York (1984)Diestel, J., Uhl, J.J.: Vector Measures. Mathematical Surveys and Monographs, vol. 15. American Mathematical Society, Providence (1977)Dieudonné, J.: Sur la convergence de suites de measures de Radon. An. Acad. Brasi. Ciên. 23, 277–282 (1951)Ferrando, J.C.: Strong barrelledness properties in certain l_{0}^{\infty }({{\fancyscript {A}}} ) l 0 ∞ ( A ) spaces. J. Math. Anal. Appl. 190, 194–202 (1995)Ferrando, J.C., López-Pellicer, M.: Strong barrelledness properties in l 0 ∞ ( X , A ) and bounded finite additive measures. Math. Ann. 287, 727–736 (1990)Kakol, J., López-Pellicer, M.: On Valdivia strong version of Nikodym boundedness property, preprintKöthe, G.: Topological Vector Spaces I and II. Springer, Berlin (1979)López-Pellicer, M.: Webs and bounded finitely additive measures. J. Math. Anal. Appl. 210, 257–267 (1997)Nikodym, O.M.: Sur les familles bornées de fonctions parfaitement additives d’ensembles abstrait. Monatsh. Math. U. Phys. 40, 418–426 (1933)Schachermayer, W.: On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras. Dissertationes Math. (Rozprawy Mat.) 214, 33 pp., 1982Valdivia, M.: On the closed graph theorem. Collect. Math. 22, 51–72 (1971)Valdivia, M.: On certain barrelled normed spaces. Ann. Inst. Fourier (Grenoble) 29, 39–56 (1979)Valdivia, M.: On Nikodym boundedness property, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 107, 355–372, 201
Completeness in the Mackey topology
Bonet and Cascales [Non-complete Mackey topologies on Banach spaces, Bulletin of the Australian Mathematical Society, 81, 3 (2010), 409-413], answering a question of M. Kunze and W. Arendt, gave an example of a norming norm-closed subspace N of the dual of a Banach space X such that mu(X, N) is not complete,where mu(X, N) denotes the Mackey topology associated with the dual pair aEuroX, NaEuro parts per thousand. We prove in this note that we can decide on the completeness or incompleteness of topologies of this form in a quite general context, thus providing large classes of counterexamples to the aforesaid question. Moreover, our examples use subspaces N of X* that contain a predual P of X (if exists), showing that the phenomenon of noncompleteness that Kunze and Arendt were looking for is not only relatively common but illustrated by "well-located" subspaces of the dual. We discuss also the situation for a typical Banach space without a predual-the space c (0)-and for the James space J.The first author is supported in part by MICINN and FEDER (project no. MTM2008-05396), by Fundacion Seneca (project no. 08848/PI/08), by Generalitat Valenciana (GV/2010/036), and by Universitat Politecnica de Valencia (project no. PAID-06-09-2829). The second author is supported in part by MICINN project no. MTM2011-22417, by Generalitat Valenciana (GV/2010/036), and by Universidad Politecnica de Valencia (project no. PAID-06-09-2829).Guirao Sánchez, AJ.; Montesinos Santalucia, V. (2015). Completeness in the Mackey topology. Functional Analysis and Its Applications. 49(2):97-105. https://doi.org/10.1007/s10688-015-0091-2S97105492J. Bonet and B. Cascales, “Non-complete Mackey topologies on Banach spaces,” Bull. Aust. Math. Soc., 81:3 (2010), 409–413.M. Fabian, P. Habala, P. Hájek, V. Montesinos, and V. Zizler, Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books in Math., Springer-Verlag, New York, 2011.P. Pérez-Carreras and J. Bonet, Barreled Locally Convex Spaces, North-Holland Mathematical Studies, vol. 131, North-Holland, Amsterdam, 1987.P. Civin and B. Yood, “Quasi-reflexive spaces,” Proc. Amer. Math. Soc., 8:5 (1957), 906–911.J. Diestel, Sequences and Series in Banach Spaces, Graduate Text in Math., vol. 92, Springer-Verlag, New York, 1984.K. Floret, Weakly Compact Sets, Lecture Notes in Math., vol. 801, Springer-Verlag, Berlin, 1980.G. Godefroy, “Boundaries of convex sets and interpolation sets,” Math. Ann., 277:2 (1987), 173–184.R. C. James, “On nonreflexive Banach space isometric with its second conjugate,” Proc. Nat. Acad. Sci. USA, 37 (1951), 174–177.G. Köthe, Topological Vector Spaces I, Springer-Verlag, New York, 1969
The impact of organ motion and the appliance of mitigation strategies on the effectiveness of hypoxia-guided proton therapy for non-small cell lung cancer.
BACKGROUND AND PURPOSE
To investigate the impact of organ motion on hypoxia-guided proton therapy treatments for non-small cell lung cancer (NSCLC) patients.
MATERIALS AND METHODS
Hypoxia PET and 4D imaging data of six NSCLC patients were used to simulate hypoxia-guided proton therapy with different motion mitigation strategies including rescanning, breath-hold, respiratory gating and tumour tracking. Motion-induced dose degradation was estimated for treatment plans with dose painting of hypoxic tumour sub-volumes at escalated dose levels. Tumour control probability (TCP) and dosimetry indices were assessed to weigh the clinical benefit of dose escalation and motion mitigation. In addition, the difference in normal tissue complication probability (NTCP) between escalated proton and photon VMAT treatments have been assessed.
RESULTS
Motion-induced dose degradation was found for target coverage (CTV V95% up to -4%) and quality of the dose-escalation-by-contour (QRMS up to 6%) as a function of motion amplitude and amount of dose escalation. The TCP benefit coming from dose escalation (+4-13%) outweighs the motion-induced losses (<2%). Significant average NTCP reductions of dose-escalated proton plans were found for lungs (-14%), oesophagus (-10%) and heart (-16%) compared to conventional VMAT plans. The best plan dosimetry was obtained with breath hold and respiratory gating with rescanning.
CONCLUSION
NSCLC affected by hypoxia appears to be a prime target for proton therapy which, by dose-escalation, allows to mitigate hypoxia-induced radio-resistance despite the sensitivity to organ motion. Furthermore, substantial reduction in normal tissue toxicity can be expected compared to conventional VMAT. Accessibility and standardization of hypoxia imaging and clinical trials are necessary to confirm these findings in a clinical setting
Out of distribution detection for intra-operative functional imaging
Multispectral optical imaging is becoming a key tool in the operating room.
Recent research has shown that machine learning algorithms can be used to
convert pixel-wise reflectance measurements to tissue parameters, such as
oxygenation. However, the accuracy of these algorithms can only be guaranteed
if the spectra acquired during surgery match the ones seen during training. It
is therefore of great interest to detect so-called out of distribution (OoD)
spectra to prevent the algorithm from presenting spurious results. In this
paper we present an information theory based approach to OoD detection based on
the widely applicable information criterion (WAIC). Our work builds upon recent
methodology related to invertible neural networks (INN). Specifically, we make
use of an ensemble of INNs as we need their tractable Jacobians in order to
compute the WAIC. Comprehensive experiments with in silico, and in vivo
multispectral imaging data indicate that our approach is well-suited for OoD
detection. Our method could thus be an important step towards reliable
functional imaging in the operating room.Comment: The final authenticated version is available online at
https://doi.org/10.1007/978-3-030-32689-0_
Normal Cones and Thompson Metric
The aim of this paper is to study the basic properties of the Thompson metric
in the general case of a real linear space ordered by a cone . We
show that has monotonicity properties which make it compatible with the
linear structure. We also prove several convexity properties of and some
results concerning the topology of , including a brief study of the
-convergence of monotone sequences. It is shown most of the results are
true without any assumption of an Archimedean-type property for . One
considers various completeness properties and one studies the relations between
them. Since is defined in the context of a generic ordered linear space,
with no need of an underlying topological structure, one expects to express its
completeness in terms of properties of the ordering, with respect to the linear
structure. This is done in this paper and, to the best of our knowledge, this
has not been done yet. The Thompson metric and order-unit (semi)norms
are strongly related and share important properties, as both are
defined in terms of the ordered linear structure. Although and
are only topological (and not metrical) equivalent on , we
prove that the completeness is a common feature. One proves the completeness of
the Thompson metric on a sequentially complete normal cone in a locally convex
space. At the end of the paper, it is shown that, in the case of a Banach
space, the normality of the cone is also necessary for the completeness of the
Thompson metric.Comment: 36 page
Compact convex sets in 2-dimensional asymmetric normed lattices
[EN] In this note, we study the geometric structure of compact convex sets in 2-dimensional asymmetric normed lattices. We prove that every q-compact convex set is strongly q-compact and we give a complete geometric description of the compact convex set with non empty interior in (R-2, q), where q is an asymmetric lattice norm.The first author has been supported by CONACYT (Mexico) under Grant 204028. The second author has been supported by the Ministerio de Economia y Competitividad (Spain) under Grant MTM2012-36740-C02-02.Jonard-Perez, N.; Sánchez Pérez, EA. (2016). Compact convex sets in 2-dimensional asymmetric normed lattices. Quaestiones Mathematicae. 39(1):73-82. https://doi.org/10.2989/16073606.2015.1023864S738239
Duals of variable exponent Hörmander spaces () and some applications
In this paper we characterize the dual \bigl(\B^c_{p(\cdot)} (\Omega)
\bigr)' of the variable exponent H\"or\-man\-der space \B^c_{p(\cdot)}
(\Omega) when the exponent satisfies the conditions , the Hardy-Littlewood maximal operator is
bounded on for some and is
an open set in . It is shown that the dual
\bigl(\B^c_{p(\cdot)} (\Omega) \bigr)' is isomorphic to the
H\"ormander space \B^{\mathrm{loc}}_\infty (\Omega) (this is the
counterpart of the isomorphism \bigl(\B^c_{p(\cdot)} (\Omega) \bigr)'
\simeq \B^{\mathrm{loc}}_{\widetilde{p'(\cdot)}} (\Omega), , recently proved by the authors) and hence the
representation theorem
\bigl( \B^c_{p(\cdot)} (\Omega) \bigr)' \simeq
l^{\N}_\infty is obtained. Our proof
relies heavily on the properties of
the Banach envelopes of the steps of \B^c_{p(\cdot)} (\Omega) and on the
extrapolation theorems in the variable Lebesgue spaces of entire
analytic functions obtained in a precedent paper. Other results for
, , are also given (e.g. \B^c_p
(\Omega) does not contain any infinite-dimensional -Banach
subspace with or the quasi-Banach space \B_p \cap
\E'(Q) contains a copy of when is a cube in ).
Finally, a question on complex interpolation (in the sense of Kalton)
of variable exponent H\"ormander spaces is proposed.J. Motos is partially supported by grant MTM2011-23164 from the Spanish Ministry of Science and Innovation. The authors wish to thank the referees for the careful reading of the manuscript and for many helpful suggestions and remarks that improved the exposition. In particular, the remark immediately following Theorem 2.1 and the Question 2 were motivated by the comments of one of them.Motos Izquierdo, J.; Planells Gilabert, MJ.; Talavera Usano, CF. (2015). Duals of variable exponent Hörmander spaces () and some applications. Revista- Real Academia de Ciencias Exactas Fisicas Y Naturales Serie a Matematicas. 109(2):657-668. https://doi.org/10.1007/s13398-014-0209-zS6576681092Aboulaich, R., Meskine, D., Souissi, A.: New diffussion models in image processing. Comput. Math. Appl. 56(4), 874–882 (2008)Acerbi, E., Mingione, G.: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164(3), 213–259 (2002)Bastero, J.: l q -subspaces of stable p -Banach spaces, 0 < p ≤ 1 . Arch. Math. (Basel) 40, 538–544 (1983)Boas, R.P.: Entire functions. Academic Press, London (1954)Boza, S.: Espacios de Hardy discretos y acotación de operadores. Dissertation, Universitat de Barcelona (1998)Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue spaces, foundations and harmonic analysis. Birkhäuser, Basel (2013)Cruz-Uribe, D.: SFO, A. Fiorenza, J. M. Martell, C. Pérez: The boundedness of classical operators on variable L p spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and sobolev spaces with variable exponents. lecture notes in mathematics, vol. 2007. Springer, Berlin, Heidelberg (2011)Hörmander, L.: The analysis of linear partial operators II, Grundlehren 257. Springer, Berlin, Heidelberg (1983)Hörmander, L.: The analysis of linear partial operators I, Grundlehren 256. Springer, Berlin, Heidelberg (1983)Kalton, N.J., Peck, N.T., Roberts, J.W.: An F -space sampler, London Mathematical Society Lecture Notes, vol. 89. Cambridge University Press, Cambridge (1985)Kalton, N.J.: Banach envelopes of non-locally convex spaces. Canad. J. Math. 38(1), 65–86 (1986)Kalton, N.J., Mitrea, M.: Stability results on interpolation scales of quasi-Banach spaces and applications. Trans. Am. Math. Soc. 350(10), 3903–3922 (1998)Kalton, N.J.: Quasi-Banach spaces, Handbook of the Geometry of Banach Spaces, vol. 2. In: Johnson, W.B., Lindenstrauss, J. (eds.), pp. 1099–1130. Elsevier, Amsterdam (2003)Köthe, G.: Topological vector spaces I. Springer, Berlin, Heidelberg (1969)Motos, J., Planells, M.J., Talavera, C.F.: On variable exponent Lebesgue spaces of entire analytic functions. J. Math. Anal. Appl. 388, 775–787 (2012)Motos, J., Planells, M.J., Talavera, C.F.: A note on variable exponent Hörmander spaces. Mediterr. J. Math. 10, 1419–1434 (2013)Stiles, W.J.: Some properties of l p , 0 < p < 1 . Studia Math. 42, 109–119 (1972)Triebel, H.: Theory of function spaces. Birkhäuser, Basel (1983)Vogt, D.: Sequence space representations of spaces of test functions and distributions. In: Zapata, G.I. (ed.) Functional analysis, holomorphy and approximation theory, Lecture Notes in Pure and Applied Mathematics, vol. 83, pp. 405–443 (1983
Twisted convolution and Moyal star product of generalized functions
We consider nuclear function spaces on which the Weyl-Heisenberg group acts
continuously and study the basic properties of the twisted convolution product
of the functions with the dual space elements. The final theorem characterizes
the corresponding algebra of convolution multipliers and shows that it contains
all sufficiently rapidly decreasing functionals in the dual space.
Consequently, we obtain a general description of the Moyal multiplier algebra
of the Fourier-transformed space. The results extend the Weyl symbol calculus
beyond the traditional framework of tempered distributions.Comment: LaTeX, 16 pages, no figure
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