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Putinar proved that hyponormal operators are sub-scalar. In this note we will use Putinar’s construction to prove that the generalized commutator \n \n \n \n $$T^*{\\hat{{\\hat{X}}}}-{\\hat{{\\hat{X}}}}S^*$$\n \n belongs to the Hilbert-Schmidt class whenever \n \n \n \n $$TX-XS$$\n \n does so with \n \n \n \n $$T,S^*$$\n \n hyponormal operators and \n \n \n \n $$\\hat{{\\hat{X}}}$$\n \n is some transform of X that depends on T and S. Also, we prove that the commutator \n \n \n \n $$T^* \\hat{{\\hat{X}}}-\\hat{{\\hat{X}}} T^*$$\n \n is Hilbert-Schmidt when T is a hyponormal operators of finite cyclic multiplicity and \n \n \n \n $$TX-XT$$\n \n is a Hilbert-Schmidt operator.",{"EN":628},"Some consequences of Putinar’s model of hyponormal operators concerning Hilbert-Schmidt commutators",{"VOID":630},"[\"5700902453391919459\"]",{"VOID":632},"Abdessemed, A., Davies, E.B.: Some commutator estimates in the Schatten classes. J. London Math. Soc. 39(2), 299–308 (1989)\nBerger, C.A., Shaw, B.I.: Selfcommutators of multicyclic hyponormal operators are always trace class. Bull. Amer. Math. Soc. 79, 1193–1199 (1973)\nVoiculescu, Dan: A note on quasitrianguliarity and trace-class self-commutators. Acta Sci. Math. (Szeged) 42(1–2), 195–199 (1980)\nHadwin, Don, Nordgren, Eric: Extensions of the Berger-Shaw theorem. Proc. Amer. Math. Soc. 102(3), 517–525 (1988)\nKittaneh, Fuad: On generalized Fuglede-Putnam theorem of Hilbert-Schmidt type. Proc. Amer. Math. Soc. 88(2), 293–298 (1983)\nGary, W.: The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators II. J. Operator Theory 5(1), 3–16 (1981)\nPutinar, Mihai: Hyponormal operators are subscalar. J. Operator Theory 12(2), 385–395 (1984)\nFuruta, Takayuki: An extension of the Fuglede-Putnam theorem to subnormal operators using a Hilbert-Schmidt norm inequality. Proc. Amer. Math. Soc. 81(2), 240–242 (1981)\nFuruta, Takayuki: On relaxation of normality in the Fuglede-Putnam theorem. Proc. Amer. Math. Soc. 77(3), 324–328 (1979)\nShulman, Victor, Turowska, Lyudmila: Operator synthesis. II. Individual synthesis and linear operator equations. J. Reine Angew. Math. 590, 143–187 (2006)\nShulman, Victor: Some remarks on the Fuglede-Weiss theorem. Bull. London Math. 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In abstract terms, letH be a Hilbert space andA: D (A)⊂H→H be an unbounded normal operator, we consider the boundary value problemü(t)=Au(t), 0\u003Ct\u003C∞,u(0)=u\n 0∈D(A),\n \n \n \n \n$$\\mathop {\\lim }\\limits_{t \\to 0} \\left\\| {u\\left( t \\right)} \\right\\| = 0$$\n\n . The problem of recoveringu\n 0 whenu(T) is known for someT>0 is not well-posed. Suppose we are given approximationsx\n 1,x\n 2,…,x\n N tou(T\n 1),…,u(T\n N) with 0\u003CT, \u003C…\u003CT\n N and positive weightsP\n i,i=1,…,n,\n \n \n \n \n$$\\sum\\limits_{i = 1}^N {P_i = 1} $$\n\n such that\n \n \n \n \n$$Q_2 \\left( {u_0 } \\right) = \\sum\\limits_{i = 1}^N {P_i } \\left\\| {u\\left( {T_i } \\right) - x_i } \\right\\|^2 \\leqslant \\varepsilon ^2 $$\n\n . If ‖u\n t(0)‖≤E for some a priori constantE, we construct a regularized solution ν(t) such that\n \n \n \n \n$$Q\\left( {\\nu \\left( 0 \\right)} \\right) \\leqslant \\varepsilon ^2 $$\n\n while\n \n \n \n \n$$\\left\\| {u\\left( 0 \\right) - \\nu \\left( 0 \\right)} \\right\\| = 0\\left( {ln \\left( {E\u002F\\varepsilon } \\right)} \\right)^{ - 1} $$\n\n and\n \n \n \n \n$$\\left\\| {u\\left( t \\right) - \\nu \\left( t \\right)} \\right\\| = 0\\left( {\\varepsilon ^{\\beta \\left( t \\right)} } \\right)$$\n\n where 0\u003Cβ(t)\u003C1 and the constant in the order symbol depends uponE. The function β(t) is larger thant\u002Fm whent\u003CT\n k andk is the largest integer such that\n \n \n \n \n$$(\\sum\\limits_{k = 1}^N {P_i (T_i )} )\u003C (\\sum\\limits_{k = 1}^N {P_i (T_i )} = m$$\n\n , which β(t)=t\u002Fm on [T\n k, m] and β(t)=1 on [m, ∞). Similar results are obtained if the measurement is made in the maximum norm, i.e.,Q\n ∞(u\n 0)=max{‖u(T\n i)−x\n i‖, 1≤i≤N}.",{"EN":711},"Estimates and regularization for solutions of some ill-posed problems of elliptic and parabolic type",{"VOID":713},"[\"11662159117856871971\"]",{"VOID":715},"10.1007\u002FBF02844893","2024-05-02T06:22:03.955+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF02844893",[719,744],{"id":720,"sortIndex":21,"researcher":20,"roles":721,"affiliations":722,"properties":739,"displayName":741,"givenName":20,"familyName":20},"30c891df-7238-49a8-b096-86174f755279",[252],[723,731],{"id":724,"sortIndex":21,"affiliation":725,"properties":20},"c8039f48-acb1-4df1-9f55-c6ab480c684c",{"id":724,"createTime":20,"updateTime":20,"relativeEntities":726,"slug":20,"properties":727,"entityType":20,"verifyStatus":20,"verifyTime":20,"verifyNote":20,"languages":20,"translateLanguages":20,"viewCount":20,"url":20,"parentIds":730,"statistic":20},[],{"title":728},{"VI":729},"Department of Mathematics, Iowa State University, Ames",[],{"id":732,"sortIndex":94,"affiliation":733,"properties":20},"bfd7d580-8966-4433-b56b-3cb2bbc79164",{"id":732,"createTime":20,"updateTime":20,"relativeEntities":734,"slug":20,"properties":735,"entityType":20,"verifyStatus":20,"verifyTime":20,"verifyNote":20,"languages":20,"translateLanguages":20,"viewCount":20,"url":20,"parentIds":738,"statistic":20},[],{"title":736},{"VI":737},"Istituto Analisi Globale e Applicazioni, Consiglio Nazionale delle Ricerche, Firenze, Italia",[],{"title":740,"gsAuthor":742},{"VI":741},"Howard A. 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this research paper, we propose a new iterative algorithm for finding a common solution to fixed point problems of demicontractive mapping and variational inequality problems which involves monotone and Lipschitz continuous operators in the framework of real Hilbert spaces. We incorporate a viscosity iterative technique, using subgradient extragradient method, we prove under standard assumptions that the iterative sequence generated from our algorithm strongly converges to the solution set, assuming the solution set is consistent. Furthermore, we adopt a self-adaptive stepsize that is being generated at each iteration, which is independent of the Lipschitz constant of the singled-valued operator. Our result is an improvement and an extension of many results in this direction.",{"EN":963},"Finding a common solution of variational inequality and fixed point problems using subgradient extragradient techniques",{"VOID":965},"[\"6553921932877838015\"]",{"VOID":967},"Abbas, M., Iqbal, H.: Two inertial extragradient viscosity algorithms for solving variational inequality and fixed point problems. J. Nonlinear Var. Anal. 4, 377–398 (2020)\nBaiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Wiley, New York (1984)\nBeck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)\nCeng, L.-C., Shang, M.: Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings. Optimization (2019). https:\u002F\u002Fdoi.org\u002F10.1080\u002F02331934.2019.1647203\nCensor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)\nCensor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Methods Softw. 26, 827–845 (2011)\nChidume, C.E., Măruşter, Ş: Iterative methods for the computation of fixed points of demicontractive mappings. J. Comput. Appl. Math. 234(3), 861–882 (2010). https:\u002F\u002Fdoi.org\u002F10.1016\u002Fj.cam.2010.01.050\nGibali, A., Jolaoso, L.O., Mewomo, O.T., Taiwo, A.: Fast and simple Bregman projection methods for solving variational inequalities and related problems in Banach spaces. Results Math. 75, Art. No. 179, 36 pp (2020)\nGibali, A.: A new non-Lipschitzian projection method for solving variational inequalities in Euclidean spaces. J. Nonlinear Anal. Optim. 6, 41–51 (2015)\nGodwin, E.C., Izuchukwu, C., Mewomo, O.T.: An inertial extrapolation method for solving generalized split feasibility problems in real Hilbert spaces. Boll. Unione Mat. Ital. 14(2), 379–401 (2021)\nGodwin, E.C., Izuchukwu, C., Mewomo, O.T.: Image restoration using a modified relaxed inertial method for generalized split feasibility problems Math. Methods Appl. Sci. 46(5), 5521–5544 (2023)\nHe, B.S., Liao, L.Z.: Improvements of some projection methods for monotone nonlinear variational inequalities. J. Optim. Theory Appl. 112, 111–128 (2002)\nHicks, T.L., Kubicek, J.D.: On the Mann iteration process in a Hilbert space. J. Math. Anal. Appl. 59(3), 498–504 (1977). https:\u002F\u002Fdoi.org\u002F10.1016\u002F0022-247x(77)90076-2\nIzuchukwu, C, Mebawondu, A.A., Mewomo, O.T.: A new method for solving split variational inequality problems without co-coerciveness. J. Fixed Point Theory Appl. 22(4), Art. No. 98 (2020)\nIzuchukwu, C., Ogwo, G.N., Mewomo, O.T.: An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions. Optimization (2020). https:\u002F\u002Fdoi.org\u002F10.1080\u002F02331934.1808648\nJolaoso, L.O., Taiwo, A., Alakoya, O.T., Mewomo, O.T.: Strong convergence theorem for solving pseudomonotone variational inequality problem using projection method in a reflexive Banach space. J. Optim. Theory Appl. 185(3), 744–766 (2020)\nKhan, S.H., Alakoya, T.O., Mewomo, O.T.: Relaxed projection methods with self-adaptive step size for solving variational inequality and fixed point problems for an infinite family of multivalued relatively nonexpansive mappings in Banach Spaces. Math. Comput. Appl. 25, Art. 54 (2020)\nKorpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody 12, 747–756 (1976)\nKraikaew, R., Saejung, S.: Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 163, 399–412 (2014)\nLiu, H., Yang, J.: Weak convergence of iterative methods for solving quasimonotone variational inequalities. Comput. Optim. Appl. 77(2), 491–508 (2020)\nLorenz, D., Pock, T.: An inertial forward–backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51, 311–325 (2015)\nLuo, Y.L., Tan, B.: A self-adaptive inertial extragradient algorithm for solving pseudo-monotone variational inequality in Hilbert spaces. J. Nonlinear Convex Anal. (in press) (2020)\nMaingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)\nMainge, P.E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325, 469–479 (2007)\nMaingé, P.E.: Convergence theorems for inertial KM-type algorithms. J. Comput. Appl. Math. 219, 223–236 (2008)\nNadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006)\nNesterov, Y.: A method of solving a convex programming problem with convergence rate O(1\u002Fk2). Sov. Math. Dokl. 27, 372–376 (1983)\nOgwo, G.N., Izuchukwu, C., Mewomo, O.T.: Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity. Numer. Algorithms 88(3), 1419–1456 (2022)\nOgwo, G.N., Izuchukwu, C., Shehu, Y., Mewomo, O.T.: Convergence of relaxed inertial subgradient extragradient methods for quasimonotone variational inequality problems. J. Sci. Comput. (2022). https:\u002F\u002Fdoi.org\u002F10.1007\u002Fs10915-021-01670-1\nOgwo, G.N., Izuchukwu, C., Mewomo, O.T.: Relaxed inertial methods for solving split variational inequality problems without product space formulation. Acta Math. Sci. Ser. B (Engl. Ed.) 42, 1701–1733 (2022)\nPolyak, B.T.: Some methods of speeding up the convergence of iteration methods. Comput. Math. Math. Phys. 4, 1–17 (1964)\nSaejung, S., Yotkaew, P.: Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal. 75, 742–750 (2012)\nShehu, Y., Iyiola, O.S., Reich, S.: A modified inertial subgradient extragradient method for solving variational inequalities. Optim. Eng. 23, 421–449 (2022). https:\u002F\u002Fdoi.org\u002F10.1007\u002Fs11081-020-09593-w\nStampacchia, G.: Variational inequalities. In: Theory and Applications of Monotone Operators. Proceedings of the NATO Advanced Study Institute, Venice, Italy, Edizioni Odersi, Gubbio, Italy, pp. 102–192 (1968)\nTan, B., Xu, S.S., Li, S.: Inertial shrinking projection algorithms for solving hierarchical variational inequality problems. J. Nonlinear Convex Anal. (in press) (2020)\nThong, D.V., Hieu, D.V.: Inertial extragradient algorithms for strongly pseudomonotone variational inequalities. J. Comput. Appl. Math. 341, 80–98 (2018)\nThong, D.V., Hieu, D.V.: Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems. Optimization 67(1), 83–102 (2017). https:\u002F\u002Fdoi.org\u002F10.1080\u002F02331934.2017.1377199\nTian, M., Tong, M.: Self-adaptive subgradient extragradient method with inertial modification for solving monotone variational inequality problems and quasi-nonexpansive fixed point problems. J. Inequal. Appl. (2019). https:\u002F\u002Fdoi.org\u002F10.1186\u002Fs13660-019-1958-1\nXu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)\nCensor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optim. J. Math. Program. Oper. Res. 61(9), 1119–1132 (2012). https:\u002F\u002Fdoi.org\u002F10.1080\u002F02331934.2010.539689\nEzeora, J.N., Francis, O.: Nwawuru: an inertial-based hybrid and shrinking projection methods for solving split common fixed point problems in real reflexive spaces. Int. J. Nonlinear Anal. Appl. 14(1), 2541–2556 (2023). https:\u002F\u002Fdoi.org\u002F10.22075\u002Fijnaa.2022.24912.2852\nNwawuru, F.O., Ezeora, J.N.: Inertial-based extragradient algorithm for approximating a common solution of split-equilibrium problems and fixed-point problems of nonexpansive semigroups. J. Inequal. Appl. 2023, 22 (2023). https:\u002F\u002Fdoi.org\u002F10.1186\u002Fs13660-023-02923-3\nEzeora, J.N., Enyi, C.D., Nwawuru, F.O., Richard, C.: Ogbonna. An algorithm for split equilibrium and fixed-point problems using inertial extragradient techniques. Comput. Appl. Math. 42, 103 (2023). https:\u002F\u002Fdoi.org\u002F10.1007\u002Fs40314-023-02244-7\nGoebel, K., Reich, S.: Uniform Convexity. 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Modelling 37(12–13), 1307–1315 (2003)","https:\u002F\u002Fwww.sciencedirect.com\u002Fscience\u002Farticle\u002Fpii\u002FS0895717703900422",{"doi":1238},"10.1016\u002Fs0895-7177(03)90042-2",{"id":1240,"createTime":1241,"updateTime":1242,"relativeEntities":1243,"slug":1244,"properties":1245,"entityType":243,"verifyStatus":244,"verifyTime":1256,"verifyNote":246,"languages":20,"translateLanguages":20,"viewCount":21,"primaryUrl":1257,"fullTextUrl":20,"authors":1258,"publicationType":278,"publisherRelationship":1289,"citationCount":21,"citationInfo":1331,"publishDate":1333,"publishYear":1146,"citationAnalyzeStatus":19,"lastCitationAnalyze":1334,"indexDatabases":1335,"openAccess":20,"references":20,"isForceReanalyzing":327},"d4ab8cf1-1c3b-4614-80fc-c4020f8ecf48","2023-12-05T06:20:39.878+00:00","2026-07-17T09:47:37.168+00:00",[],"Existence-of-renormalized-solutions-for-nonlocal-thermistor-problem-via-weak-convergence-of-truncations",{"abstract":1246,"title":1248,"gsPaper":1250,"references":1252,"doi":1254},{"EN":1247},"A thermistor is an electric circuit device made of ceramic material whose electric conductivity depends on the temperature. The mathematical model of this device takes the form of a system that consists of a nonlinear parabolic equation describing the temperature. In this paper, we investigate the model in the presence of a nonlocal term in the right hand side of the equation describing the evolution of the temperature. Precisely, we establish the existence of a renormalized solution to the problem (2) by using the weak convergence of a considered truncation.",{"EN":1249},"Existence of renormalized solutions for nonlocal thermistor problem via weak convergence of truncations",{"VOID":1251},"[\"17460121173026583546\"]",{"VOID":1253},"Aberqi, A., Bennouna, J., Hammoumi, M.: Uniqueness of renormalized solutions for a class of parabolic equations. Ric 66(2), 629–644 (2017)\nAntontsev, S.N., Chipot, M.: The thermistor problem: existence, smoothness uniqueness, blowup. SIAM J. Math. Anal. 25(4), 1128–1156 (1994). (SIAM)\nBénilan, P., Boccardo, L., Gallouët, Thierry, Gariepy, R., Pierre, M., Vázquez, Juan L.: An \\(L^{1}\\)-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. della Sc. Norm. Super. di Pisa-Classe di Sci. 22(2), 241–273 (1995)\nBlanchard, D., Francfort, G.: Study of a doubly nonlinear heat equation with no growth assumptions on the parabolic term. SIAM J. Math. Anal. 19(5), 1032–1056 (1988). (SIAM)\nBlanchard, D., Murat, F.: Renormalised solutions of nonlinear parabolic problems with \\(L^1\\) data: existence and uniqueness. Proc. R. Soc. Edinb. A: Math. 127(6), 1137–1152 (1997). (Royal Society of Edinburgh Scotland Foundation)\nBlanchard, D., Murat, F., Redwane, H.: Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems. Differ. Equ. 177(2), 331–374 (2001). (Academic Press)\nBlanchard, D., Redwane, H.: Renormalized solutions for a class of nonlinear evolution problems. J. Math. Pures Appl. 77(2), 117–151 (1998). (Elsevier)\nBlanchard, D.: Truncations and monotonicity methods for parabolic equations. Nonlinear Anal. Theory Methods 21(10), 725–743 (1993). (Elsevier)\nBoccardo, L., Giachetti, D., Diaz, J. I., Murat, F.: Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms. J. Differ. Equ. 106(2), 215–237 (1993)\nBourahma, M., Benkirane, A., Bennouna, J.: Existence of renormalized solutions for some nonlinear elliptic equations in Orlicz spaces. Rend. Circ. Mat. Palermo 69(1), 231–252 (2020)\nDall’Aglio, A., Orsina, L.: Nonlinear parabolic equations with natural growth conditions and L1 data. Nonlinear Anal. Theory Methods 27(1), 59–73 (1996). (Elsevier)\nDiPerna, R.J., Lions, P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 321–366, JSTOR (1989)\nDiPerna, R.J., Lions, P.-L.: On the Fokker-Planck-Boltzmann equation. Commun. Math. Phys. 120(1), 1–23 (1988). (Springer)\nDiaz, J.I., MURAT, F.: Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms. J. Diff. Eq. 106, 215–237 (1993)\nEl Hachimi, A., Sidi Ammi M.R., Torres, Delfim FM.: Existence and uniqueness of solutions for a nonlocal parabolic thermistor-type problem, arXiv math\u002F0512629 (2005)\nEl Hachimi, A., Ammi, M.R., Sidi.: Existence of weak solutions for the thermistor problem with degeneracy. Electron. J. Differ. Equ.: 127–137, p. 2002. Southwest Texas State University, Department of Mathematics, San Marcos, TX (2002)\nGlitzky, A., Liero, M., Nika, G.: Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discr. Cont. DYN-S 14(11), 3953 (2021)\nGrenon, N.: Resultats d’existence et comportement asymptotique pour des equations paraboliques quasi-lineaires. Orléans (1990)\nLacey, A.A.: Thermal runaway in a non-local problem modelling Ohmic heating: part I: model derivation and some special cases. Eur. J. Appl. Math. 6(2), 127–144 (1995). (Cambridge University Press)\nKouraichi, C., Siai, A.: Equivalence between entropy and renormalized solutions for parabolic equations. Indag. Math. 26(4), 679–696 (2015)\nLacey, A.A.: Thermal runaway in a non-local problem modelling Ohmic heating. Part II: general proof of blow-up and asymptotics of runaway. Eur. J. Appl. Math. 6(3), 201–224 (1995). (Cambridge University Press)\nLandes, R.: On the existence of weak solutions for quasilinear parabolic initial-boundary value problems. Proc. R. Soc. Edinb. A: Math. 89(3–4), 217–237 (1981). (Royal Society of Edinburgh Scotland Foundation)\nLions, P.-L.: Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models, 2. Oxford University Press on Demand (1996)\nMurat, F.: Soluciones renormalizadas de EDP elipticas no lineales, preprint, 93023, Citeseer (1993)\nMurat, F.: Équations elliptiques non linéaires avec second membre L1 ou mesure. Actes du. 26, A12–A24 (1994)\nRedwane, H.: Existence of a solution for a class of parabolic equations with three unbounded nonlinearities. Adv. Dyn. Syst. Appl. 2(2), 241–264 (2007). (Citeseer)\nOuédraogo, A., Houede, D.A., Ibrango, I.: Renormalized solutions for convection-diffusion problems involving a nonlocal operator. Nonlinear Differ. Equs. Appl. NoDEA 28(5), 1–27 (2021)\nSimon, J.: Compact sets in the Lp (0, T; B) spaces. Ann. Mat. Pura Appl. 4(146), 65–96 (1987)\nVorob’yov, N.A., Mukminov, F.K.: Existence of a renormalized solution of a parabolic problem in anisotropic Sobolev-Orlicz spaces. J. Math. 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