Trace ideals for Fourier integral operators with non-smooth symbols II

We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We establish continuity and Schatten-von Neumann properties of such operators when acting on modulation spaces.Comment: 25 page

Catecholaminergic signalling through thymic nerve fibres, thymocytes and stromal cells is dependent on both circulating and locally synthesized glucocorticoids

Glucocorticoids have been shown to modulate the expression of noradrenaline metabolizing enzymes and beta(2)- and alpha(1B)-adrenoceptors in a tissue- and cell- specific manner. In the thymus, apart from extensive sympathetic innervation, a regulatory network has been identified that encompasses catecholamine-containing non-lymphoid and lymphoid cells. We examined a putative role of adrenal- and thymus-derived glucocorticoids in modulation of rat thymic noradrenaline levels and adrenoceptor expression. Seven days postadrenalectomy, the thymic levels of mRNAs encoding tyrosine hydroxylase, dopamine beta-hydroxylase, monoamine oxidase-A and, consequently, noradrenaline were decreased. Catecholamine content was diminished in autofluorescent nerve fibres (judging by the intensity of fluorescence) and thymocytes (considering HPLC measurements of noradrenaline and the frequency of tyrosine hydroxylase-positive cells), while it remained unaltered in non-lymphoid autofluorescent cells. In addition, adrenalectomy diminished the thymocyte expression of beta(2)- and alpha(1B)-adrenoceptors at both mRNA and protein levels. Administration of ketoconazole (an inhibitor of glucocorticoid synthesis/action; 25 mg kg(-1) day(-1), s.c.) to glucocorticoid-deprived rats increased the thymic levels of tyrosine hydroxylase, dopamine beta-hydroxylase and, consequently, noradrenaline. The increased intensity of the autofluorescent cell fluorescence in ketoconazole-treated rats indicated an increase in their catecholamine content, and suggested differential glucocorticoid-mediated regulation of catecholamines in thymic lymphoid and non-lymphoid cells. In addition, ketoconazole increased the thymocyte expression of alpha(1B)-adrenoceptors. Thus, this study indicates that in the thymus, as in some other tissues, glucocorticoids not only act in concert with cateholamines, but they may modulate catecholamine action by tuning thymic catecholamine metabolism and adrenoceptor expression in a cell-specific manner. Additionally, the study indicates a role of thymus-derived glucocorticoids in this modulation

Variational Problems with Fractional Derivatives: Euler-Lagrange Equations

We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these two bounds coincide, we derive a new form of Euler-Lagrange equations. We use approximations for fractional derivatives in the Lagrangian and obtain the Euler-Lagrange equations which approximate the initial Euler-Lagrange equations in a weak sense

Revitalizacija i informaciona integracija proizvodnih resursa u cilju podizanja konkurentnosti TRAYAL korporacije na međunarodnom tržištu – rekapitulacija ukupnih rezultata na projektu TR-6362A

U uvodnom delu ovog rada izloženi su organizacija, osnovni sadržaj planiranih istraživanja i ciljevi na projektu TR-6362A koji su zajednički realizovali Mašinski fakultet Univerziteta u Beogradu, Trayal Korporacija iz Kruševca i Informatika A.D. iz Beograda. Dalje se detaljno navode ključni rezultati projekta ostvareni u trogodišnjem istraživačkom periodu koji su realizovani sa globalnim ciljem sistematskog unapređenja stanja izabranih proizvodnih resursa Trayal korporacije za domen mehaničke prerade elastomera, upravljanja proizvodnjom i praćenja kvaliteta proizvoda. Pored toga u radu se navode i neka opšta zapažanja vezana za stanje Trayal korporacije i faze kroz koje je prolazila u procesu svojinske transformacije. Na kraju, navode se dalje perspektive praktične primene ostvarenih rezultata i mogućnosti nastavka saradnje po ovom osnovu u narednom periodu

Revitalizacija i informaciona integracija proizvodnih resursa u cilju podizanja konkurentnosti TRAYAL korporacije na međunarodnom tržištu – ciljevi, rezultati i planirane istraživačko-razvojne aktivnosti na projektu TR-6362A

U okviru ovog rada izloženi su organizacija, sadržaj planiranih istraživanja i ciljevi na projektu TR-6362A koji zajednički realizuju Mašinski fakultet Univerziteta u Beogradu, Trayal Korporacija iz Kruševca i Informatika AD iz Beograda. Takođe, u radu se navode rezultati istraživačkih aktivnosti sprovedenih u toku 2005 godine i sa tim povezana iskustva koja se primenjuju u tekućim istraživačkim aktivnostima sa ciljem podizanja nivoa praktične primenljivosti ovih istraživanja na unapređenje tekućeg stanja proizvodnih resursa Trayal korporacije. Pored toga u radu se navode i neka opšta razmatranja stanja tehnologije u domaćoj industriji prerade gume i procesa privatizacije, koji je strateško opredeljenje Vlade Republike Srbije. Navode se neka iskustva koja postoje u transformaciji gumarske industrije Ruske federacije i zemalja bivšeg SSSR-a, sa kojima postoji intenzivna saradnja firme Informatika u delu modernizacije i implementacije novih tehnologija

Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis

[EN] We use techniques from time-frequency analysis to show that the space S(omega )of rapidly decreasing omega-ultradifferentiable functions is nuclear for every weight function omega(t) = o(t) as t tends to infinity. Moreover, we prove that, for a sequence (M-p)(p) satisfying the classical condition (M1) of Komatsu, the space of Beurling type S-(M)p when defined with L-2 norms is nuclear exactly when condition (M2)' of Komatsu holds.We thank the reviewer very much for the careful reading of our manuscript and the comments to improve the paper. The first three authors were partially supported by the Project FFABR 2017 (MIUR), and by the Projects FIR 2018 and FAR 2018 (University of Ferrara). The first and third authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilita e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The research of the second author was partially supported by the project MTM2016-76647-P and the grant BEST/2019/172 from Generalitat Valenciana. The fourth author is supported by FWF-project J 3948-N35.Boiti, C.; Jornet Casanova, D.; Oliaro, A.; Schindl, G. (2021). Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis. Collectanea mathematica. 72(2):423-442. https://doi.org/10.1007/s13348-020-00296-0S423442722Asensio, V., Jornet, D.: Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(4), 3477–3512 (2019)Aubry, J.-M.: Ultrarapidly decreasing ultradifferentiable functions, Wigner distributions and density matrices. J. London Math. Soc. 2(78), 392–406 (2008)Björck, G.: Linear partial differential operators and generalized distributions. Ark. Mat. 6(21), 351–407 (1966)Boiti, C., Jornet, D., Oliaro, A.: Regularity of partial differential operators in ultradifferentiable spaces and Wigner type transforms. J. Math. Anal. Appl. 446, 920–944 (2017)Boiti, C., Jornet, D., Oliaro, A.: The Gabor wave front set in spaces of ultradifferentiable functions. Monatsh. Math. 188(2), 199–246 (2019)Boiti, C., Jornet, D., Oliaro, A.: About the nuclearity of $$\cal{S}_{(M_{p})}$$ and $$\cal{S}_{\omega }$$. In: Boggiatto, P., et al. (eds.) Advances in Microlocal and Time-Frequency Analysis. Applied and Numerical Harmonic Analysis, pp. 121–129. Birkhäuser, Cham (2020)Boiti, C., Jornet, D., Oliaro, A.: Real Paley-Wiener theorems in spaces of ultradifferentiable functions. J. Funct. Anal. 278(4), 108348 (2020)Bonet, J., Meise, R., Melikhov, S.N.: A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14(3), 425–444 (2007)Braun, R.W., Meise, R., Taylor, B.A.: Ultradifferentiable functions and Fourier analysis. Result. Math. 17, 206–237 (1990)Fernández, C., Galbis, A., Jornet, D.: Pseudodifferential operators on non-quasianalytic classes of Beurling type. Studia Math. 167(2), 99–131 (2005)Fernández, C., Galbis, A., Jornet, D.: Pseudodifferential operators of Beurling type and the wave front set. J. Math. Anal. Appl. 340(2), 1153–1170 (2008)Franken, U.: Weight functions for classes of ultradifferentiable functions. Results Math. 25, 50–53 (1994)Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)Gröchenig, K., Leinert, M.: Wiener’s Lemma for twisted convolution and Gabor frames. J. Am. Math. Soc. 17(1), 1–18 (2004)Gröchenig, K., Zimmermann, G.: Spaces of Test Functions via the STFT. J. Funct. Spaces Appl. 2(1), 25–53 (2004)Heinrich, T., Meise, R.: A support theorem for quasianalytic functionals. Math. Nachr. 280(4), 364–387 (2007)Hörmander, L.: Notions of Convexity. Progress in Mathematics, vol. 127. Birkhäuser, Boston (1994)Janssen, A.J.E.M.: Duality and Biorthogonality for Weyl-Heisenberg Frames. J. Fourier Anal. Appl. 1(4), 403–436 (1995)Komatsu, H.: Ultradistributions I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect IA Math. 20, 25–105 (1973)Langenbruch, M.: Hermite functions and weighted spaces of generalized functions. Manuscripta Math. 119(3), 269–285 (2006)Meise, R., Vogt, D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)Petzsche, H.J.: Die nuklearität der ultradistributionsräume und der satz vom kern I. Manuscripta Math. 24, 133–171 (1978)Pietsch, A.: Nuclear Locally Convex Spaces. Springer, Berlin (1972)Pilipović, S., Prangoski, B., Vindas, J.: On quasianalytic classes of Gelfand-Shilov type. Parametrix and convolution. J. Math. Pures Appl. 116, 174–210 (2018)Rodino, L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific Publishing Co. Inc, River Edge, NJ (1993)Rodino, L., Wahlberg, P.: The Gabor wave front set. Monatsh. Math. 173, 625–655 (2014)Schmets, J., Valdivia, M.: Analytic extension of ultradifferentiable Whitney jets. Collect. Math. 50(1), 73–94 (1999

Generalized Hamilton's Principle with Fractional Derivatives

We generalize Hamilton's principle with fractional derivatives in Lagrangian L(t,y(t),{}_0D_t^\al y(t),\alpha) so that the function $y$ and the order of fractional derivative $\alpha$ are varied in the minimization procedure. We derive stationarity conditions and discuss them through several examples

Glucocorticoids, master modulators of the thymic catecholaminergic system?

There is evidence that the major mediators of stress, i.e., catecholamines and glucocorticoids, play an important role in modulating thymopoiesis and consequently immune responses. Furthermore, there are data suggesting that glucocorticoids influence catecholamine action. Therefore, to assess the putative relevance of glucocorticoid-catecholamine interplay in the modulation of thymopoiesis we analyzed thymocyte differentiation/maturation in non-adrenalectomized and andrenalectomized rats subjected to treatment with propranolol (0.4 mg.100 g body weight(-1).day(-1)) for 4 days. The effects of beta-adrenoceptor blockade on thymopoiesis in non-adrenalectomized rats differed not only quantitatively but also qualitatively from those in adrenalectomized rats. In adrenalectomized rats, besides a more efficient thymopoiesis [judged by a more pronounced increase in the relative proportion of the most mature single-positive TCR alpha beta(high) thymocytes as revealed by two-way ANOVA; for CD4(+)CD8(-)F (1,20) = 10.92, P lt 0.01; for CD4(-)CD8(+)F (1,20) = 7.47, P lt 0.05], a skewed thymocyte maturation towards the CD4(-)CD8(+) phenotype, and consequently a diminished CD4(+)CD8(-)/CD4(-)CD8(+) mature TCR alpha beta(high) thymocyte ratio (3.41 +/- 0.21 in non-adrenalectomized rats vs 2.90 +/- 0.31 in adrenalectomized rats, P lt 0.05) were found. Therefore, we assumed that catecholaminergic modulation of thymopoiesis exhibits a substantial degree of glucocorticoid-dependent plasticity. Given that glucocorticoids, apart from catecholamine synthesis, influence adrenoceptor expression, we also hypothesized that the lack of adrenal glucocorticoids affected not only beta-adrenoceptor- but also alpha-adrenoceptor-mediated modulation of thymopoiesis

The Gabor wave front set in spaces of ultradifferentiable functions

[EN] We consider the spaces of ultradifferentiable functions S as introduced by Bjorck (and its dual S) and we use time-frequency analysis to define a suitable wave front set in this setting and obtain several applications: global regularity properties of pseudodifferential operators of infinite order and the micro-pseudolocal behaviour of partial differential operators with polynomial coefficients and of localization operators with symbols of exponential growth. Moreover, we prove that the new wave front set, defined in terms of the Gabor transform, can be described using only Gabor frames. Finally, some examples show the convenience of the use of weight functions to describe more precisely the global regularity of (ultra)distributions.The authors were partially supported by the INdAM-Gnampa Project 2016 "Nuove prospettive nell'analisi microlocale e tempo-frequenza", by FAR2013, FAR2014 (University of Ferrara) and by the project "Ricerca Locale - Analisi di Gabor, operatori pseudodifferenziali ed equazioni differenziali" (University of Torino). The research of the second author was partially supported by the project MTM2016-76647-P.Boiti, C.; Jornet Casanova, D.; Oliaro, A. (2019). The Gabor wave front set in spaces of ultradifferentiable functions. Monatshefte für Mathematik. 188(2):199-246. https://doi.org/10.1007/s00605-018-1242-3S1992461882Albanese, A., Jornet, D., Oliaro, A.: Quasianalytic wave front sets for solutions of linear partial differential operators. Integr. Equ. Oper. Theory 66, 153–181 (2010)Albanese, A., Jornet, D., Oliaro, A.: Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non-quasianalytic classes. Math. Nachr. 285(4), 411–425 (2012)Björck, G.: Linear partial differential operators and generalized distributions. Ark. Mat. 6(21), 351–407 (1966)Boiti, C., Gallucci, E.: The overdetermined Cauchy problem for $$\omega$$ ω -ultradifferentiable functions. Manuscripta Math. 155(3-4), 419–448 (2018)Boiti, C., Jornet, D.: A simple proof of Kotake–Narasimhan theorem in some classes of ultradifferentiable functions. J. Pseudo-Differ. Oper. Appl. 8(2), 297–317 (2017)Boiti, C., Jornet, D.: A characterization of the wave front set defined by the iterates of an operator with constant coefficients. Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM 111(3), 891–919 (2017)Boiti, C., Jornet, D., Juan-Huguet, J.: Wave front sets with respect to the iterates of an operator with constant coefficients. Abstr. Appl. Anal. 2014, 1–17 (2014). https://doi.org/10.1155/2014/438716Boiti, C., Jornet, D., Oliaro, A.: Regularity of partial differential operators in ultradifferentiable spaces and Wigner type transforms. J. Math. Anal. Appl. 446, 920–944 (2017)Bonet, J., Meise, R., Melikhov, S.N.: A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14(3), 425–444 (2007)Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization. Theory and Examples, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2006)Braun, R.W., Meise, R., Taylor, B.A.: Ultradifferentiable functions and Fourier analysis. Result. Math. 17, 206–237 (1990)Cappiello, M., Schulz, R.: Microlocal analysis of quasianalytic Gelfand–Shilov type ultradistributions. Complex Var. Elliptic Equ. 61(4), 538–561 (2016)Carypis, E., Wahlberg, P.: Propagation of exponential phase space singularities for Schrödinger equations with quadratic Hamiltonians. J. Fourier Anal. Appl. 23(3), 530–571 (2017)Christensen, O.: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Springer, Berlin (2016)Fernández, C., Galbis, A., Jornet, D.: Pseudodifferential operators on non-quasianalytic classes of Beurling type. Studia Math. 167(2), 99–131 (2005)Fernández, C., Galbis, A., Jornet, D.: Pseudodifferential operators of Beurling type and the wave front set. J. Math. Anal. Appl. 340(2), 1153–1170 (2008)Fieker, C.: $$P$$ P -Konvexität und $$\omega$$ ω -Hypoelliptizität für partielle Differentialoperatoren mit konstanten Koeffizienten. Diplomarbeit, Mathematischen Institut der Heinrich-Heine-Universität Düsseldorf (1993)Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)Gröchenig, K., Zimmermann, G.: Spaces of test functions via the STFT. J. Funct. Spaces Appl. 2(1), 25–53 (2004)Heil, C.: A Basis Theory Primer. Applied and Numerical Harmonic Analysis. Springer, New York (2011)Hörmander, L.: Fourier integral operators. Acta Math. 127(1), 79–183 (1971)Hörmander, L.: Quadratic hyperbolic operators. In: Cattabriga, L., Rodino, L. (eds.) Microlocal Analysis and Applications. Lecture Notes in Mathematics, pp. 118–160. Springer, Berlin (1991)Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. I. Springer-Verlag, Berlin (1983)Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. II. Springer-Verlag, Berlin (1983)Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. III. Springer-Verlag, Berlin (1985)Janssen, A.J.E.M.: Duality and biorthogonality for Weyl–Heisenberg frames. J. Fourier Anal. Appl. 1(4), 403–436 (1995)Langenbruch, M.: Hermite functions and weighted spaces of generalized functions. Manuscripta Math. 119(3), 269–285 (2006)Meise, R., Vogt, D.: Introduction to Functional Analysis. Oxford Science Publications, Clarendon Press, Oxford (1997)Nakamura, S.: Propagation of the homogeneous wave front set for Schrödinger equations. Duke Math. J. 126, 349–367 (2005)Nicola, F., Rodino, L.: Global Pseudo-Differential Calculus on Euclidean Spaces. Springer, Basel (2010)Pilipović, S.: Tempered ultradistributions. Boll. U.M.I. B (7) 2(2), 235-251 (1988)Prangoski, B.: Pseudodifferential operators of infinite order in spaces of tempered ultradistributions. J. Pseudo-Differ. Oper. Appl. 4(4), 495–549 (2013)Pilipović, S., Prangoski, B.: Anti-Wick and Weyl quantization on ultradistribution spaces. J. Math. Pures Appl. 103(2), 472–503 (2015)Rodino, L.: Linear Partial Differential Operators and Gevrey Spaces. World Scientific Publishing Co., Inc., River Edge, NJ (1993)Rodino, L., Wahlberg, P.: The Gabor wave front set. Monatsh. Math. 173, 625–655 (2014)Schulz, R., Wahlberg, P.: Microlocal properties of Shubin pseudodifferential and localization operators. J. Pseudo-Differ. Oper. Appl. 7(1), 91–111 (2016)Schulz, R., Wahlberg, P.: Equality of the homogeneous and the Gabor wave front set. Commun. Partial Differ. Equ. 42(5), 703–730 (2017)Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer-Verlag, Berlin (1987)Sjöstrand, J.: Singularités analytiques microlocales. Astérisque 95, 1–166 (1982)Toft, J.: The Bargmann transform on modulation and Gelfand–Shilov spaces, with applications to Toeplitz and pseudo-differential operators. J. Pseudo-Differ. Oper. Appl. 3(2), 145–227 (2012)Toft, J.: Images of function and distribution spaces under the Bargmann transform. J. Pseudo-Differ. Oper. Appl. 8(1), 83–139 (2017)Treves, F.: Topological vector spaces, distributions and kernels. Academic Press, New York (1967

GM-CSF-Producing Th Cells in Rats Sensitive and Resistant to Experimental Autoimmune Encephalomyelitis

Given that granulocyte macrophage colony-stimulating factor (GM-CSF) is identified as the key factor to endow auto-reactive Th cells with the potential to induce neuroinflammation in experimental autoimmune encephalomyelitis (EAE) models, the frequency and phenotype of GM-CSF-producing (GM-CSF+) Th cells in draining lymph nodes (dLNs) and spinal cord (SC) of Albino Oxford (AO) and Dark Agouti (DA) rats immunized for EAE were examined. The generation of neuroantigen-specific GM-CSF+ Th lymphocytes was impaired in dLNs of AO rats (relatively resistant to EAE induction) compared with their DA counterparts (susceptible to EAE) reflecting impaired CD4+ lymphocyte proliferation and less supportive of GM-CSF+ Th cell differentiation dLN cytokine microenvironment. Immunophenotyping of GM-CSF+ Th cells showed their phenotypic heterogeneity in both strains and revealed lower frequency of IL-17+ IFN-gamma+, IL-17+ IFN-gamma-, and IL-17-IFN-gamma+ cells accompanied by higher frequency of IL-17-IFN-gamma- cells among them in AO than in DA rats. Compared with DA, in AO rats was also found (i) slightly lower surface density of CCR2 (drives accumulation of highly pathogenic GM-CSF+ IFN-gamma+ Th17 cells in SC) on GM-CSF+ IFN-gamma+ Th17 lymphocytes from dLNs, and (ii) diminished CCL2 mRNA expression in SC tissue, suggesting their impaired migration into the SC. Moreover, dLN and SC cytokine environments in AO rats were shown to be less supportive of GM-CSF+ IFN-gamma+ Th17 cell differentiation (judging by lower expression of mRNAs for IL-1 beta, IL-6 and IL-23/p19). In accordance with the (i) lower frequency of GM-CSF+ Th cells in dLNs and SC of AO rats and their lower GM-CSF production, and (ii) impaired CCL2 expression in the SC tissue, the proportion of proinflammatory monocytes among peripheral blood cells and their progeny (CD45(hi) cells) among the SC CD11b+ cells were reduced in AO compared with DA rats. Collectively, the results indicate that the strain specificities in efficacy of several mechanisms controlling (auto) reactive CD4+ lymphocyte expansion/differentiation into the cells with pathogenic phenotype and migration of the latter to the SC contribute to AO rat resistance to EAE