Journal articles: 'Matrice multiplicative de Hilbert'

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Haase, Gundolf, Manfred Liebmann, and Gernot Plank. "A Hilbert-order multiplication scheme for unstructured sparse matrices." International Journal of Parallel, Emergent and Distributed Systems 22, no. 4 (August 2007): 213–20. http://dx.doi.org/10.1080/17445760601122084.

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Brevig, Ole Fredrik, Karl-Mikael Perfekt, Kristian Seip, Aristomenis G. Siskakis, and Dragan Vukotić. "The multiplicative Hilbert matrix." Advances in Mathematics 302 (October 2016): 410–32. http://dx.doi.org/10.1016/j.aim.2016.07.019.

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Shparlinski, Igor E. "Multiplicative Properties of Hilbert Cubes." SIAM Journal on Discrete Mathematics 36, no. 2 (April 19, 2022): 1064–70. http://dx.doi.org/10.1137/22m1470396.

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Gerasoulis, A. "A fast algorithm for the multiplication of generalized Hilbert matrices with vectors." Mathematics of Computation 50, no. 181 (January 1, 1988): 179. http://dx.doi.org/10.1090/s0025-5718-1988-0917825-9.

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Hegyvári, Norbert. "On additive and multiplicative Hilbert cubes." Journal of Combinatorial Theory, Series A 115, no. 2 (February 2008): 354–60. http://dx.doi.org/10.1016/j.jcta.2007.05.005.

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LONG, YINXIANG, DAOWEN QIU, and DONGYANG LONG. "AN EFFICIENT SEPARABILITY CRITERION FOR n-PARTITE ARBITRARILY DIMENSIONAL QUANTUM STATES." International Journal of Quantum Information 09, no. 04 (June 2011): 1101–12. http://dx.doi.org/10.1142/s0219749911007514.

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In this paper, we obtain an efficient separability criterion for bipartite quantum pure state systems, which is based on the two-order minors of the coefficient matrix corresponding to quantum state. Then, we generalize this criterion to multipartite arbitrarily dimensional pure states. Our criterion is directly built upon coefficient matrices, but not density matrices or observables, so it has the advantage of being computed easily. Indeed, to judge separability for an arbitrary n-partite pure state in a d-dimensional Hilbert space, it only needs at most O(d) times operations of multiplication and comparison. Our criterion can be extended to mixed states. Compared with Yu's criteria, our methods are faster, and can be applied to any quantum state.

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Crane, Daniel K., and Mark S. Gockenbach. "The Singular Value Expansion for Arbitrary Bounded Linear Operators." Mathematics 8, no. 8 (August 12, 2020): 1346. http://dx.doi.org/10.3390/math8081346.

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The singular value decomposition (SVD) is a basic tool for analyzing matrices. Regarding a general matrix as defining a linear operator and choosing appropriate orthonormal bases for the domain and co-domain allows the operator to be represented as multiplication by a diagonal matrix. It is well known that the SVD extends naturally to a compact linear operator mapping one Hilbert space to another; the resulting representation is known as the singular value expansion (SVE). It is less well known that a general bounded linear operator defined on Hilbert spaces also has a singular value expansion. This SVE allows a simple analysis of a variety of questions about the operator, such as whether it defines a well-posed linear operator equation and how to regularize the equation when it is not well posed.

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DOPLICHER, SERGIO, CLAUDIA PINZARI, and JOHN E. ROBERTS. "AN ALGEBRAIC DUALITY THEORY FOR MULTIPLICATIVE UNITARIES." International Journal of Mathematics 12, no. 04 (June 2001): 415–59. http://dx.doi.org/10.1142/s0129167x01000770.

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Multiplicative unitaries are described in terms of a pair of commuting shifts of relative depth two. They can be generated from ambidextrous Hilbert spaces in a tensor C*-category. The algebraic analogue of the Takesaki–Tatsuuma Duality Theorem characterizes abstractly C*-algebras acted on by unital endomorphisms that are intrinsically related to the regular representation of a multiplicative unitary. The relevant C*-algebras turn out to be simple and indeed separable if the corresponding multiplicative unitaries act on a separable Hilbert space. A categorical analogue provides internal characterizations of minimal representation categories of a multiplicative unitary. Endomorphisms of the Cuntz algebra related algebraically to the grading are discussed as is the notion of braided symmetry in a tensor C*-category.

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STITT, JOSEPH P., and KARL M. NEWELL. "THE MULTIPLICATIVE NOISE COMPONENTS OF ISOMETRIC FORCE FLUCTUATIONS." Fluctuation and Noise Letters 12, no. 04 (December 2013): 1350026. http://dx.doi.org/10.1142/s0219477513500260.

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The experiment tested the relative contribution of additive and multiplicative noise components in isometric force fluctuations. We implemented the Hilbert–Huang transform to analyze variability in the steady state region of isometric force time series generated by participants in three adult age categories: (1) young (20–24 yrs); (2) old (60–69 yrs); and older-old (75–90 yrs). Each participant generated isometric force output in response to constant target force levels (5–40% maximum voluntary contraction). The empirical mode decomposition component of the Hilbert–Huang transform was applied to each isometric force time series resulting in a set of intrinsic mode functions (IMFS). The amplitude modulation envelopes of the IMFS obtained from the modulus of the Hilbert transform were random sequences. These random sequences were the source of the multiplicative noise. Analyses showed significant age group differences in the multiplicative noise structure of the force output. The results provide preliminary evidence that multiplicative noise dominates additive noise in isometric force output and can discriminate among the force fluctuations of healthy adult participants of different ages.

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Perfekt, Karl-Mikael, and Alexander Pushnitski. "On the spectrum of the multiplicative Hilbert matrix." Arkiv för Matematik 56, no. 1 (2018): 163–83. http://dx.doi.org/10.4310/arkiv.2018.v56.n1.a10.

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Babenko, V. F., and N. A. Kryachko. "Inequalities of Hardy-Littlewood-Polya type for operators in Hilbert space." Researches in Mathematics 21 (August 11, 2013): 34. http://dx.doi.org/10.15421/241304.

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Courtney, Dennis, and Donald Sarason. "A mini-max problem for self-adjoint Toeplitz matrices." MATHEMATICA SCANDINAVICA 110, no. 1 (March 1, 2012): 82. http://dx.doi.org/10.7146/math.scand.a-15198.

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We study a minimum problem and associated maximum problem for finite, complex, self-adjoint Toeplitz matrices. If $A$ is such a matrix, of size $(N+1)$-by-$(N+1)$, we identify $A$ with the operator it represents on ${\mathcal P}_N$, the space of complex polynomials of degrees at most $N$, with the usual Hilbert space structure it inherits as a subspace of $L^2$ of the unit circle. The operator $A$ is the compression to ${\mathcal P}_N$ of the multiplication operator on $L^2$ induced by any function in $L^{\infty}$ whose Fourier coefficients of indices between $-N$ and $N$ match the matrix entries of $A$. Our minimum problem is to minimize the $L^{\infty}$ norm of such inducers. We show there is a unique one of minimum norm, and we describe it. The associated maximum problem asks for the maximum of the ratio of the preceding minimum to the operator norm. That problem remains largely open. We present some suggestive numerical evidence.

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Noethen, Florian. "Computing Covariant Lyapunov Vectors in Hilbert spaces." Journal of Computational Dynamics 8, no. 3 (2021): 325. http://dx.doi.org/10.3934/jcd.2021014.

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<p style='text-indent:20px;'>Covariant Lyapunov Vectors (CLVs) are intrinsic modes that describe long-term linear perturbations of solutions of dynamical systems. With recent advances in the context of semi-invertible multiplicative ergodic theorems, existence of CLVs has been proved for various infinite-dimensional scenarios. Possible applications include the derivation of coherent structures via transfer operators or the stability analysis of linear perturbations in models of increasingly higher resolutions.</p><p style='text-indent:20px;'>We generalize the concept of Ginelli's algorithm to compute CLVs in Hilbert spaces. Our main result is a convergence theorem in the setting of [<xref ref-type="bibr" rid="b19">19</xref>]. The theorem relates the speed of convergence to the spectral gap between Lyapunov exponents. While the theorem is restricted to the above setting, our proof requires only basic properties that are given in many other versions of the multiplicative ergodic theorem.</p>

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Toft, Joachim, Anupam Gumber, Ramesh Manna, and P. K. Ratnakumar. "Translation and modulation invariant Hilbert spaces." Monatshefte für Mathematik 196, no. 2 (July 19, 2021): 389–98. http://dx.doi.org/10.1007/s00605-021-01589-7.

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AbstractLet $$\mathcal H$$ H be a Hilbert space of distributions on $$\mathbf{R}^{d}$$ R d which contains at least one non-zero element of the Feichtinger algebra $$S_0$$ S 0 and is continuously embedded in $$\mathscr {D}'$$ D ′ . If $$\mathcal H$$ H is translation and modulation invariant, also in the sense of its norm, then we prove that $$\mathcal H= L^2$$ H = L 2 , with the same norm apart from a multiplicative constant.

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15

Englert, Berthold-Georg, and Heng Huat Chan. "Multiplicative functions arising from the study of mutually unbiased bases." New Zealand Journal of Mathematics 51 (August 12, 2021): 65–78. http://dx.doi.org/10.53733/99.

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We introduce two families of multiplicative functions, which generalize the somewhat unusual function that was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics. In addition, we report yet another multiplicative function, which is also suggested by that example; it can be used to express the squarefree part of an integer in terms of an exponential sum.

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Alpay, Daniel, Palle Jorgensen, and Motke Porat. "White noise space analysis and multiplicative change of measures." Journal of Mathematical Physics 63, no. 4 (April 1, 2022): 042102. http://dx.doi.org/10.1063/5.0042756.

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In this paper, we display a family of Gaussian processes, with explicit formulas and transforms. This is presented with the use of duality tools in such a way that the corresponding path-space measures are mutually singular. We make use of a corresponding family of representations of the canonical commutation relations (CCR) in an infinite number of degrees of freedom. A key feature of our construction is explicit formulas for associated transforms; these are infinite-dimensional analogs of Fourier transforms. Our framework is that of Gaussian Hilbert spaces, reproducing kernel Hilbert spaces and Fock spaces. The latter forms the setting for our CCR representations. We further show, with the use of representation theory and infinite-dimensional analysis, that our pairwise inequivalent probability spaces (for the Gaussian processes) correspond in an explicit manner to pairwise disjoint CCR representations.

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Huang, Norden E., Kun Hu, Albert C. C. Yang, Hsing-Chih Chang, Deng Jia, Wei-Kuang Liang, Jia Rong Yeh, et al. "On Holo-Hilbert spectral analysis: a full informational spectral representation for nonlinear and non-stationary data." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 374, no. 2065 (April 13, 2016): 20150206. http://dx.doi.org/10.1098/rsta.2015.0206.

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The Holo-Hilbert spectral analysis (HHSA) method is introduced to cure the deficiencies of traditional spectral analysis and to give a full informational representation of nonlinear and non-stationary data. It uses a nested empirical mode decomposition and Hilbert–Huang transform (HHT) approach to identify intrinsic amplitude and frequency modulations often present in nonlinear systems. Comparisons are first made with traditional spectrum analysis, which usually achieved its results through convolutional integral transforms based on additive expansions of an a priori determined basis, mostly under linear and stationary assumptions. Thus, for non-stationary processes, the best one could do historically was to use the time–frequency representations, in which the amplitude (or energy density) variation is still represented in terms of time. For nonlinear processes, the data can have both amplitude and frequency modulations (intra-mode and inter-mode) generated by two different mechanisms: linear additive or nonlinear multiplicative processes. As all existing spectral analysis methods are based on additive expansions, either a priori or adaptive, none of them could possibly represent the multiplicative processes. While the earlier adaptive HHT spectral analysis approach could accommodate the intra-wave nonlinearity quite remarkably, it remained that any inter-wave nonlinear multiplicative mechanisms that include cross-scale coupling and phase-lock modulations were left untreated. To resolve the multiplicative processes issue, additional dimensions in the spectrum result are needed to account for the variations in both the amplitude and frequency modulations simultaneously. HHSA accommodates all the processes: additive and multiplicative, intra-mode and inter-mode, stationary and non-stationary, linear and nonlinear interactions. The Holo prefix in HHSA denotes a multiple dimensional representation with both additive and multiplicative capabilities.

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18

Nikolov-Radenkovic, Jovana. "Some additive and multiplicative results for generalized inverses." Filomat 29, no. 9 (2015): 2049–57. http://dx.doi.org/10.2298/fil1509049n.

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In this paper we give necessary and sufficient conditions for A1{1,3} + A2{1, 3}+ ... + Ak{1,3} ? (A1 + A2 + ... + Ak){1,3} and A1{1,4} + A2{1,4} + ... + Ak{1,4} ? (A1 + A2 + ... + Ak){1,4} for regular operators on Hilbert space. We also consider similar inclusions for {1,2,3}- and {1,2,4}-i inverses. We give some new results concerning the reverse order law for reflexive generalized inverses.

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Neguţ, Andrei, Georg Oberdieck, and Qizheng Yin. "Motivic decompositions for the Hilbert scheme of points of a K3 surface." Journal für die reine und angewandte Mathematik (Crelles Journal) 2021, no. 778 (April 19, 2021): 65–95. http://dx.doi.org/10.1515/crelle-2021-0015.

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Abstract We construct an explicit, multiplicative Chow–Künneth decomposition for the Hilbert scheme of points of a K3 surface. We further refine this decomposition with respect to the action of the Looijenga–Lunts–Verbitsky Lie algebra.

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Duncan, T. E., B. Maslowski, and B. Pasik-Duncan. "Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise." Stochastic Processes and their Applications 115, no. 8 (August 2005): 1357–83. http://dx.doi.org/10.1016/j.spa.2005.03.011.

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PINZARI, C., and J. E. ROBERTS. "REGULAR OBJECTS, MULTIPLICATIVE UNITARIES AND CONJUGATION." International Journal of Mathematics 13, no. 06 (August 2002): 625–65. http://dx.doi.org/10.1142/s0129167x02001423.

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The notion of left (respectively right) regular object of a tensor C*-category equipped with a faithful tensor functor into the category of Hilbert spaces is introduced. If such a category has a left (respectively right) regular object, it can be interpreted as a category of corepresentations (respectively representations) of some multiplicative unitary. A regular object is an object of the category which is at the same time left and right regular in a coherent way. A category with a regular object is endowed with an associated standard braided symmetry. Conjugation is discussed in the context of multiplicative unitaries and their associated Hopf C*-algebras. It is shown that the conjugate of a left regular object is a right regular object in the same category. Furthermore the representation category of a locally compact quantum group has a conjugation. The associated multiplicative unitary is a regular object in that category.

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Nagolkina, Zoya, and Yuri Filonov. "SCHEME OF MULTIPLICATIVE REPRESENTATIONS FOR A STOCHASTIC EQUATION WITH TWO WIENER PROCESSES." APPLIED GEOMETRY AND ENGINEERING GRAPHICS, no. 102 (June 27, 2022): 136–48. http://dx.doi.org/10.32347/0131-579x.2022.102.136-148.

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In this paper we consider a stochastic differential equation with two independent Wiener processes in an infinite-dimensional Hilbert space. This equation can be a mathematical model of a dynamic system in the presence of several independent perturbing random factors. To study the parameters of this equation, the Daletsky-Trotter method of multiplicative representations is used. This method is applied to both deterministic and stochastic equations. The method consists in the following: one constructs a partition of the interval of existence of the solution [t0, T] into elementary [tk+1, tk]. On each elementary segment, the evolutionary resolving operator of the complete equation S(tk+1, tk) is considered, as well as the product of the resolving operators of equations that are fragments of the complete equation =Q1k×S1k×S2k Thus, two multiplicative families consisting of different resolving operators are compared. When the equivalence conditions, which are verified in this paper, are satisfied, it can be argued that the solution of a stochastic equation can, in a certain sense, be represented as a composition of the corresponding solutions of differential equations on elementary intervals, the right parts of which are drift, and, accordingly, diffusion. Moreover, in order to implement such a multiplicative scheme in the case of several independent Wiener processes, additional requirements regarding diffusion coefficients should be imposed. Namely: the diffusion coefficients must be commuting operators, continuous in time. The scheme of multiplicative representations is based on the study of the parameters of evolutionary families of decision operators, as well as their estimates in the norms of the corresponding spaces. In this case, to obtain a certain estimate, several iteration steps are considered for the corresponding equations in the Hilbert space. It should be noted that the scheme of multiplicative representations can be interpreted as a scheme for obtaining an approximate solution.

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Kaliszewski, S., and John Quigg. "Equivariance and imprimivity for discrete Hopf C*-coactions." Bulletin of the Australian Mathematical Society 62, no. 2 (October 2000): 253–72. http://dx.doi.org/10.1017/s0004972700018736.

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Let U, V, and W be multiplicative unitaries coming from discrete Kac systems such that W is an amenable normal submultiplicative unitary of V with quotient U. We define notions for right-Hilbert bimodules of coactions of SV and ŜV, their restrictions to SW and ŜU, their dual coactions, and their full and reduced crossed products. If N (A) denotes the imprimitivity bimodule associated to a coaction δ of SV on a C*-algebra A by Ng's imprimitivity theorem, we prove that for a suitably nondegenerate injective right-Hilbert bimodule coaction of SV on AXB, the balanced tensor products and are isomorphic right-Hilbert A×ŜV×rSU − B × ŜW bimodules. This can be interpreted as a natural equivalence between certain crossed-product functors.

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Paterson, Alan L. T. "The Fourier Algebra for Locally Compact Groupoids." Canadian Journal of Mathematics 56, no. 6 (December 1, 2004): 1259–89. http://dx.doi.org/10.4153/cjm-2004-055-8.

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AbstractWe introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.

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Du, Fapeng, and Zuhair Nashed. "Additive perturbations and multiplicative perturbations for the core inverse of bounded linear operator in Hilbert space." Filomat 32, no. 17 (2018): 6131–44. http://dx.doi.org/10.2298/fil1817131d.

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In this paper, we present some characteristics and expressions of the core inverse A# of bounded linear operator A in Hilbert spaces. Additive perturbations of core inverse are investigated under the condition R( ?)?N(A#) = {0} and an upper bound of ||?#-A#|| is obtained. We also discuss the multiplicative perturbations. The expressions of core inverse of perturbed operator T = EAF and the upper bounds of ||T#-A#|| are obtained too.

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HUANG, NORDEN E., XIANYAO CHEN, MEN-TZUNG LO, and ZHAOHUA WU. "ON HILBERT SPECTRAL REPRESENTATION: A TRUE TIME-FREQUENCY REPRESENTATION FOR NONLINEAR AND NONSTATIONARY DATA." Advances in Adaptive Data Analysis 03, no. 01n02 (April 2011): 63–93. http://dx.doi.org/10.1142/s1793536911000659.

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As the original definition on Hilbert spectrum was given in terms of total energy and amplitude, there is a mismatch between the Hilbert spectrum and the traditional Fourier spectrum, which is defined in terms of energy density. Rigorous definitions of Hilbert energy and amplitude spectra are given in terms of energy and amplitude density in the time-frequency space. Unlike Fourier spectral analysis, where the resolution is fixed once the data length and sampling rate is given, the time-frequency resolution could be arbitrarily assigned in Hilbert spectral analysis (HSA). Furthermore, HSA could also provide zooming ability for detailed examination of the data in a specific frequency range with all the resolution power. These complications have made the conversion between Hilbert and Fourier spectral results difficult and the conversion formula is elusive until now. We have derived a simple relationship between them in this paper. The conversion factor turns out to be simply the sampling rate for the full resolution cases. In case of zooming, there is another additional multiplicative factor. The conversion factors have been tested in various cases including white noise, delta function, and signals from natural phenomena. With the introduction of this conversion, we can compare HSA and Fourier spectral analysis results quantitatively.

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Nagolkina, Zoya, and Yuri Filonov. "MULTIPLICATIVE APPROXIMATION OF A RANDOM PROCESS." APPLIED GEOMETRY AND ENGINEERING GRAPHICS, no. 100 (May 24, 2021): 205–14. http://dx.doi.org/10.32347/0131-579x.2021.100.205-214.

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In this paper we consider the stochastic Ito differential equation in an infinite-dimensional real Hilbert space. Using the method of multiplicative representations of Daletsky - Trotter, its approximate solution is constructed. Under classical conditions on the coefficients, there is a single to the stochastic equivalence of solutions of the stochastic equation, which is a random process. This development generates an evolutionary family of resolving operators by the formula x(t)= S(t, Construct the division of the segment by the points. An equation with time-uniform coefficients is considered on each elementary segment . There is a single solution of this equation on the elementary segment, which generates the resolving operator by the formula The multiplicative expression is constructed. Using the method of Dalecki-Trotter multiplicative representations, it is proved that this multiplicative expression is stochastically equivalent to the representation generated by the solution of the original equation. This means that the specified multiplicative expression is respectively a representation of the solution of the original equation. That is, the probability of one coincides with the solution of the original stochastic equation. It should be noted that this is possible under additional conditions for the coefficients of the equation. These conditions are the time continuity of the coefficients of the equation. Thus, the constructed multiplicative representation can be interpreted as an approximate solution of the original equation. This method of multiplicative approximation makes it possible to simplify the study of the corresponding random process both at the elementary segment and as a whole. It is known, that the solution of a stochastic equation by a known formula generates a solution of the inverse Kolmogorov equation in the corresponding space. This scheme of multiplicative approximation can be transferred to the solution of the parabolic equation, which is the inverse Kolmogorov equation. Thus, the method of multiplicative approximation makes it possible to simplify the study of both stochastic equations and partial differential equations.

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Radjavi, Heydar. "Sublinearity and Other Spectral Conditions on a Semigroup." Canadian Journal of Mathematics 52, no. 1 (February 1, 2000): 197–224. http://dx.doi.org/10.4153/cjm-2000-009-5.

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AbstractSubadditivity, sublinearity, submultiplicativity, and other conditions are considered for spectra of pairs of operators on a Hilbert space. Sublinearity, for example, is a weakening of the well-known property L and means σ(A + λB) ⊆ σ(A) + λσ(B) for all scalars λ. The effect of these conditions is examined on commutativity, reducibility, and triangularizability of multiplicative semigroups of operators. A sample result is that sublinearity of spectra implies simultaneous triangularizability for a semigroup of compact operators.

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Akhtari, Fatemeh, and Rasoul Nasr-Isfahani. "Amenability properties of unitary co-representations of locally compact quantum groups." International Journal of Mathematics 30, no. 14 (December 2019): 1950077. http://dx.doi.org/10.1142/s0129167x19500770.

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For locally compact quantum groups [Formula: see text], we initiate an investigation of stable states with respect to unitary co-representations [Formula: see text] of [Formula: see text] on Hilbert spaces [Formula: see text]; in particular, we study the subject on the multiplicative unitary operator [Formula: see text] of [Formula: see text] with some examples on locally compact quantum groups arising from discrete groups and compact groups. As the main result, we consider the one co-dimensional Hilbert subspace of [Formula: see text] associated to a suitable vector [Formula: see text], to present an operator theoretic characterization of stable states with respect to a related unitary co-representation [Formula: see text]. This provides a quantum version of an interesting result on unitary representations of locally compact groups given by Lau and Paterson in 1991.

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Zheleznyak, Аlexander V. "Multiplicative property of series used in the Nevanlinna-Pick problem." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 9, no. 1 (2022): 37–45. http://dx.doi.org/10.21638/spbu01.2022.104.

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In the paper we obtained substantially new sufficient condition for negativity of coefficients of power series inverse to series with positive ones. It has been proved that element-wise product of power series retains this property. In particular, it gives rise to generalization of the classical Hardy theorem about power series. These results are generalized for cases of series with multiple variables. Such results are useful in Nevanlinna – Pick theory. For example, if function k(x, y) can be represented as power series Pn≥0 an(x¯y)n, an > 0, and reciprocal function 1/k(x, y) can be represented as power series Pn≥0 bn(x¯y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna – Pick property. The reproducing kernel 1/(1 − x¯y) of the classical Hardy space H2(D) is a prime example for our theorems.

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Moradi, Hamid, and Mohammad Sababheh. "New estimates for the numerical radius." Filomat 35, no. 14 (2021): 4957–62. http://dx.doi.org/10.2298/fil2114957m.

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In this article, we present new inequalities for the numerical radius of the sum of two Hilbert space operators. These new inequalities will enable us to obtain many generalizations and refinements of some well known inequalities, including multiplicative behavior of the numerical radius and norm bounds. Among many other applications, it is shown that if T is accretive-dissipative, then 1/?2 ||T|| ? ?(T), where ?(?) and ||?||denote the numerical radius and the usual operator norm, respectively. This inequality provides a considerable refinement of the well known inequality 1/2 ||T|| ? ?(T).

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Wang, Feng-Yu. "Gradient estimates and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift." Journal of Differential Equations 260, no. 3 (February 2016): 2792–829. http://dx.doi.org/10.1016/j.jde.2015.10.020.

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DELBOURGO, DANIEL. "EXCEPTIONAL ZEROES OF P-ADIC L-FUNCTIONS OVER NON-ABELIAN FIELD EXTENSIONS." Glasgow Mathematical Journal 58, no. 2 (July 21, 2015): 385–432. http://dx.doi.org/10.1017/s0017089515000245.

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AbstractSuppose E is an elliptic curve over $\Bbb Q$, and p>3 is a split multiplicative prime for E. Let q ≠ p be an auxiliary prime, and fix an integer m coprime to pq. We prove the generalised Mazur–Tate–Teitelbaum conjecture for E at the prime p, over number fields $K\subset \Bbb Q\big(\mu_{{q^{\infty}}},\;\!^{q^{\infty}\!\!\!\!}\sqrt{m}\big)$ such that p remains inert in $K\cap\Bbb Q(\mu_{{q^{\infty}}})^+$. The proof makes use of an improved p-adic L-function, which can be associated to the Rankin convolution of two Hilbert modular forms of unequal parallel weight.

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Ungureanu, Viorica Mariela. "Stability, stabilizability and detectability for Markov jump discrete-time linear systems with multiplicative noise in Hilbert spaces." Optimization 63, no. 11 (December 6, 2012): 1689–712. http://dx.doi.org/10.1080/02331934.2012.730049.

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Longstaff, W. E., and H. Radjavi. "On Permutability and Submultiplicativity of Spectral Radius." Canadian Journal of Mathematics 47, no. 5 (October 1, 1995): 1007–22. http://dx.doi.org/10.4153/cjm-1995-053-x.

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AbstractLet r(T) denote the spectral radius of the operator T acting on a complex Hilbert space H. Let S be a multiplicative semigroup of operators on H. We say that r is permutable on 𝓢 if r(ABC) = r(BAC), for every A,B,C ∈ 𝓢. We say that r is submultiplicative on 𝓢 if r(AB) ≤ r(A)r(B), for every A, B ∈ 𝓢. It is known that, if r is permutable on 𝓢, then it is submultiplicative. We show that the converse holds in each of the following cases: (i) 𝓢 consists of compact operators (ii) 𝓢 consists of normal operators (iii) 𝓢 is generated by orthogonal projections.

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Gudrun Kalmbach HE. "Atomic kernels as waves and catastrophes." International Journal of Biological and Pharmaceutical Sciences Archive 1, no. 2 (March 30, 2021): 062–67. http://dx.doi.org/10.30574/ijbpsa.2021.1.2.0020.

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The presentation of atomic kerrnels as particles requires for the physics duality principle that they get a wave description. This is due to presenting the SU (3) GellMann matrix space by octonians which are obtained by doubling the spacetime quaternions. Their multiplication table is different from the SU (3) matrices. The third presentation of this space is a complex 4-dimensional space where the real spacetime coordinates of a 4-dimensional Euclidean Hilbert space R4 are extended to C4. For getting from Deuteron Cooper pairs NP of a neutron and proton atomic kernels AK, the wave package superpositions for AK need the mass defect of neutrons where kg is changed to inner speeds or interaction energies. For kg octonians have a GF measuring base triple as Gleason operator. Using a projective geometrical norming, C4 is normed to CP³, a projective 3-dimensional space. Its cell C³ has spacetime coordinates C², extended by an Einstein energy plane z3 = (m,f), m mass, f = 1/∆t frequency where mass can be transformed into f by using mc² = hf. If C³ is presented as a real space R6, it can be real projective normed to a real projective space P5 for the field presentation of AK. As field the NP‘s have then a common group speed for AK wave packages superpositions with which AK moves in spacetime C² and also a presentation as a Ψ wave. As probability distribution where they can be energetically found in space serves Ψ* Ψ.

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Bakherad, Mojtaba, and Mubariz T. Garayev. "Berezin number inequalities for operators." Concrete Operators 6, no. 1 (January 1, 2019): 33–43. http://dx.doi.org/10.1515/conop-2019-0003.

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Abstract The Berezin transform Ã of an operator A, acting on the reproducing kernel Hilbert space ℋ = ℋ (Ω) over some (non-empty) set Ω, is defined by Ã(λ) = 〉Aǩ λ, ǩ λ〈 (λ ∈ Ω), where ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over k} _\lambda } = {{{k_\lambda }} \over {\left\| {{k_\lambda }} \right\|}}$ is the normalized reproducing kernel of ℋ. The Berezin number of an operator A is defined by ${\bf{ber}}{\rm{(}}A) = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\tilde A(\lambda )} \right| = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\left\langle {A{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over k} }_\lambda },{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over k} }_\lambda }} \right\rangle } \right|$ . In this paper, we prove some Berezin number inequalities. Among other inequalities, it is shown that if A, B, X are bounded linear operators on a Hilbert space ℋ, then $${\bf{ber}}(AX \pm XA) \leqslant {\bf{be}}{{\bf{r}}^{{1 \over 2}}}\left( {A*A + AA*} \right){\bf{be}}{{\bf{r}}^{{1 \over 2}}}\left( {X*X + XX*} \right)$$ and $${\bf{be}}{{\bf{r}}^2}({A^*}XB) \leqslant {\left\| X \right\|^2}{\bf{ber}}({A^*}A){\bf{ber}}({B^*}B).$$ We also prove the multiplicative inequality $${\bf{ber}}(AB){\bf{ber}}(A){\bf{ber}}(B)$$

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Marinelli, Carlo, and Luca Scarpa. "Fréchet differentiability of mild solutions to SPDEs with respect to the initial datum." Journal of Evolution Equations 20, no. 3 (October 30, 2019): 1093–130. http://dx.doi.org/10.1007/s00028-019-00546-0.

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Abstract We establish n-th-order Fréchet differentiability with respect to the initial datum of mild solutions to a class of jump diffusions in Hilbert spaces. In particular, the coefficients are Lipschitz-continuous, but their derivatives of order higher than one can grow polynomially, and the (multiplicative) noise sources are a cylindrical Wiener process and a quasi-left-continuous integer-valued random measure. As preliminary steps, we prove well-posedness in the mild sense for this class of equations, as well as first-order Gâteaux differentiability of their solutions with respect to the initial datum, extending previous results by Marinelli, Prévôt, and Röckner in several ways. The differentiability results obtained here are a fundamental step to construct classical solutions to non-local Kolmogorov equations with sufficiently regular coefficients by probabilistic means.

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Matache, Mihaela T., and Valentin Matache. "Operator self-similar processes on Banach spaces." Journal of Applied Mathematics and Stochastic Analysis 2006 (May 4, 2006): 1–18. http://dx.doi.org/10.1155/jamsa/2006/82838.

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Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determined multiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such a power-function, provided that the linear space spanned by the expectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS processes whose expectation-function is differentiable on an interval (s0,∞), for some s0≥1, have a unique scaling family of operators of the form {sH:s>0}, if the expectations of the process span a dense linear subspace of category 2. The existence of a scaling family of the form {sH:s>0} is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.

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Landsman, N. P. "Poisson Spaces with a Transition Probability." Reviews in Mathematical Physics 09, no. 01 (January 1997): 29–57. http://dx.doi.org/10.1142/s0129055x97000038.

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The common structure of the space of pure states ℘ of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function p:℘×℘→[0,1], with certain properties. The Poisson structure is connected with the transition probabilities through unitarity (in a specific formulation intrinsic to the given context). In classical mechanics, where p(ρ,σ)=δρσ, unitarity poses no restriction on the Poisson structure. Quantum mechanics is characterized by a specific (complex Hilbert space) form of p, and by the property that the irreducible components of ℘ as a transition probability space coincide with the symplectic leaves of ℘ as a Poisson space. In conjunction, these stipulations determine the Poisson structure of quantum mechanics up to a multiplicative constant (identified with Planck's constant). Motivated by E. M. Alfsen, H. Hanche-Olsen and F. W. Shultz (Acta Math.144 (1980) 267–305) and F.W. Shultz (Commun. Math. Phys.82 (1982) 497–509), we give axioms guaranteeing that ℘ is the space of pure states of a unital C*-algebra. We give an explicit construction of this algebra from ℘.

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Hausenblas, Erika, and Mihály Kovács. "Global solutions to stochastic Volterra equations driven by Lévy noise." Fractional Calculus and Applied Analysis 21, no. 5 (October 25, 2018): 1170–202. http://dx.doi.org/10.1515/fca-2018-0064.

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Abstract In this paper we investigate the existence and uniqueness of semilinear stochastic Volterra equations driven by multiplicative Lévy noise of pure jump type. In particular, we consider the equation $$\begin{array}{} \left\{ \begin{aligned} du(t) & = \left( A\int_0 ^t b(t-s) u(s)\,ds\right) \, dt + F(t,u(t))\,dt \\ & {} + \int_ZG(t,u(t), z) \tilde \eta(dz,dt) + \int_{Z_L}G_L(t,u(t), z) \eta_L(dz,dt),\, t\in (0,T],\\ u(0)&=u_0, \end{aligned} \right. \end{array} $$ where Z and ZL are Banach spaces, η̃ is a time-homogeneous compensated Poisson random measure on Z with intensity measure ν (capturing the small jumps), and ηL is a time-homogeneous Poisson random measure on ZL independent to η̃ with finite intensity measure νL (capturing the large jumps). Here, A is a selfadjoint operator on a Hilbert space H, b is a scalar memory function and F, G and GL are nonlinear mappings. We provide conditions on b, F G and GL under which a unique global solution exists. We also present an example from the theory of linear viscoelasticity where our result is applicable. The specific kernel b(t) = cρtρ−2, 1 < ρ < 2, corresponds to a fractional-in-time stochastic equation and the nonlinear maps F and G can include fractional powers of A.

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NEFF, PATRIZIO, KRZYSZTOF CHEŁMIŃSKI, and HANS-DIETER ALBER. "NOTES ON STRAIN GRADIENT PLASTICITY: FINITE STRAIN COVARIANT MODELLING AND GLOBAL EXISTENCE IN THE INFINITESIMAL RATE-INDEPENDENT CASE." Mathematical Models and Methods in Applied Sciences 19, no. 02 (February 2009): 307–46. http://dx.doi.org/10.1142/s0218202509003449.

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We propose a model of finite strain gradient plasticity including phenomenological Prager type linear kinematical hardening and nonlocal kinematical hardening due to dislocation interaction. Based on the multiplicative decomposition, a thermodynamically admissible flow rule for Fp is described involving as plastic gradient Curl Fp. The formulation is covariant w.r.t. superposed rigid rotations of the reference, intermediate and spatial configuration but the model is not spin-free due to the nonlocal dislocation interaction and cannot be reduced to a dependence on the plastic metric [Formula: see text]. The linearization leads to a thermodynamically admissible model of infinitesimal plasticity involving only the Curl of the nonsymmetric plastic distortion p. Linearized spatial and material covariance under constant infinitesimal rotations is satisfied. Uniqueness of strong solutions of the infinitesimal model is obtained if two non-classical boundary conditions on the plastic distortion p are introduced: [Formula: see text] on the microscopically hard boundary ΓD ⊂ ∂Ω and [ Curl p] · τ = 0 on the microscopically free boundary ∂Ω\ΓD, where τ are the tangential vectors at the boundary ∂Ω. A weak reformulation of the infinitesimal model allows for a global in-time solution of the rate-independent initial boundary value problem. The method is based on a mixed variational inequality with symmetric and coercive bilinear form. We use a new Hilbert-space suitable for dislocation density dependent plasticity.

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Zhu, Shaolong, Maoni Chao, Jinyu Zhang, Xinjuan Xu, Puwen Song, Jinlong Zhang, and Zhongwen Huang. "Identification of Soybean Seed Varieties Based on Hyperspectral Imaging Technology." Sensors 19, no. 23 (November 28, 2019): 5225. http://dx.doi.org/10.3390/s19235225.

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Hyperspectral imaging is a nondestructive testing technology that integrates spectroscopy and iconology technologies, which enables us to quickly obtain both internal and external information of objects and identify crop seed varieties. First, the hyperspectral images of ten soybean seed varieties were collected and the reflectance was obtained. Savitzky-Golay smoothing (SG), first derivative (FD), standard normal variate (SNV), fast Fourier transform (FFT), Hilbert transform (HT), and multiplicative scatter correction (MSC) spectral reflectance pretreatment methods were used. Then, the feature wavelengths and feature information of the pretreated spectral reflectance data were extracted using competitive adaptive reweighted sampling (CARS), the successive projections algorithm (SPA), and principal component analysis (PCA). Finally, 5 classifiers, Bayes, support vector machine (SVM), k-nearest neighbor (KNN), ensemble learning (EL), and artificial neural network (ANN), were used to identify seed varieties. The results showed that MSC-CARS-EL had the highest accuracy among the 90 combinations, with training set, test set, and 5-fold cross-validation accuracies of 100%, 100%, and 99.8%, respectively. Moreover, the contribution of spectral pretreatment to discrimination accuracy was higher than those of feature extraction and classifier selection. Pretreatment methods determined the range of the identification accuracy, feature-selective methods and classifiers only changed within this range. The experimental results provide a good reference for the identification of other crop seed varieties.

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Bengulescu, Marc, Philippe Blanc, and Lucien Wald. "On the intrinsic timescales of temporal variability in measurements of the surface solar radiation." Nonlinear Processes in Geophysics 25, no. 1 (January 26, 2018): 19–37. http://dx.doi.org/10.5194/npg-25-19-2018.

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Abstract. This study is concerned with the intrinsic temporal scales of the variability in the surface solar irradiance (SSI). The data consist of decennial time series of daily means of the SSI obtained from high-quality measurements of the broadband solar radiation impinging on a horizontal plane at ground level, issued from different Baseline Surface Radiation Network (BSRN) ground stations around the world. First, embedded oscillations sorted in terms of increasing timescales of the data are extracted by empirical mode decomposition (EMD). Next, Hilbert spectral analysis is applied to obtain an amplitude-modulation–frequency-modulation (AM–FM) representation of the data. The time-varying nature of the characteristic timescales of variability, along with the variations in the signal intensity, are thus revealed. A novel, adaptive null hypothesis based on the general statistical characteristics of noise is employed in order to discriminate between the different features of the data, those that have a deterministic origin and those being realizations of various stochastic processes. The data have a significant spectral peak corresponding to the yearly variability cycle and feature quasi-stochastic high-frequency variability components, irrespective of the geographical location or of the local climate. Moreover, the amplitude of this latter feature is shown to be modulated by variations in the yearly cycle, which is indicative of nonlinear multiplicative cross-scale couplings. The study has possible implications on the modeling and the forecast of the surface solar radiation, by clearly discriminating the deterministic from the quasi-stochastic character of the data, at different local timescales.

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Arnon, Avron. "Implication, Equivalence, and Negation." Logical Investigations 27, no. 1 (May 27, 2021): 31–45. http://dx.doi.org/10.21146/2074-1472-2021-27-1-31-45.

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A system $HCL_{\overset{\neg}{\leftrightarrow}}$ in the language of {$ \neg, \leftrightarrow $} is obtained by adding a single negation-less axiom schema to $HLL_{\overset{\neg}{\leftrightarrow}}$ (the standard Hilbert-type system for multiplicative linear logic without propositional constants), and changing $ \rightarrow $ to $\leftrightarrow$. $HCL_{\overset{\neg}{\leftrightarrow}}$ is weakly, but not strongly, sound and complete for ${\bf CL}_{\overset{\neg}{\leftrightarrow}}$ (the {$ \neg,\leftrightarrow$} – fragment of classical logic). By adding the Ex Falso rule to $HCL_{\overset{\neg}{\leftrightarrow}}$ we get a system with is strongly sound and complete for ${\bf CL}_ {\overset{\neg}{\leftrightarrow}}$ . It is shown that the use of a new rule cannot be replaced by the addition of axiom schemas. A simple semantics for which $HCL_{\overset{\neg}{\leftrightarrow}}$ itself is strongly sound and complete is given. It is also shown that $L_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ , the logic induced by $HCL_{\overset{\neg}{\leftrightarrow}}$ , has a single non-trivial proper axiomatic extension, that this extension and ${\bf CL}_{\overset{\neg}{\leftrightarrow}}$ are the only proper extensions in the language of { $\neg$, $\leftrightarrow$ } of $ {\bf L}_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ , and that $ {\bf L}_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ and its single axiomatic extension are the only logics in {$ \neg, \leftrightarrow$ } which have a connective with the relevant deduction property, but are not equivalent $\neg$ to an axiomatic extension of ${\bf R}_{\overset{\neg}{\leftrightarrow}}$ (the intensional fragment of the relevant logic ${\bf R}$). Finally, we discuss the question whether $ {\bf L}_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ can be taken as a paraconsistent logic.

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Konevskikh, Tatiana, Rozalia Lukacs, Reinhold Blümel, Arkadi Ponossov, and Achim Kohler. "Mie scatter corrections in single cell infrared microspectroscopy." Faraday Discussions 187 (2016): 235–57. http://dx.doi.org/10.1039/c5fd00171d.

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Strong Mie scattering signatures hamper the chemical interpretation and multivariate analysis of the infrared microscopy spectra of single cells and tissues. During recent years, several numerical Mie scatter correction algorithms for the infrared spectroscopy of single cells have been published. In the paper at hand, we critically reviewed existing algorithms for the correction of Mie scattering and suggest improvements. We developed an iterative algorithm based on Extended Multiplicative Scatter Correction (EMSC), for the retrieval of pure absorbance spectra from highly distorted infrared spectra of single cells. The new algorithm uses the van de Hulst approximation formula for the extinction efficiency employing a complex refractive index. The iterative algorithm involves the establishment of an EMSC meta-model. While existing iterative algorithms for the correction of resonant Mie scattering employ three independent parameters for establishing a meta-model, we could decrease the number of parameters from three to two independent parameters, which reduced the calculation time for the Mie scattering curves for the iterative EMSC meta-model by a factor of 10. Moreover, by employing the Hilbert transform for evaluating the Kramers–Kronig relations based on a FFT algorithm in Matlab, we further improved the speed of the algorithm by a factor of 100. For testing the algorithm we simulate distorted apparent absorbance spectra by utilizing the exact theory for the scattering of infrared light at absorbing spheres, taking into account the high numerical aperture of infrared microscopes employed for the analysis of single cells and tissues. In addition, the algorithm was applied to measured absorbance spectra of single lung cancer cells.

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Wu, Yuqing, and Isao Noda. "Extension of Quadrature Orthogonal Signal Corrected Two-Dimensional (QOSC 2D) Correlation Spectroscopy I: Principal Component Analysis Based QOSC 2D." Applied Spectroscopy 61, no. 10 (October 2007): 1040–44. http://dx.doi.org/10.1366/000370207782217761.

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The present study proposes a new quadrature orthogonal signal correlation (QOSC) filtering method based on principal component analysis (PCA). The external perturbation variable vector typically used in the QOSC operation is replaced with a matrix consisting of the spectral data principal components (PCs) and their quadrature counterparts obtained by using the discrete Hilbert–Noda transformation. Thus, QOSC operation can be carried out for a dataset without the explicit knowledge of the external variables information. The PCA-based QOSC filtering can be most effectively applied to two-dimensional (2D) correlation analysis. The performance of this filtering operation on the simulated spectra data set with the interference of strong random noise demonstrated that the PCA-based QOSC filtering not only eliminates the influence of signals that are unrelated to the final target but also preserves the out-of-phase information in the data matrix essential for asynchronous correlation analysis. The result of 2D correlation analysis has also demonstrated that essentially only one principal component is necessary for PCA-based QOSC to perform well. Although the present PCA-based QOSC filtering scheme is not as powerful as that based on the explicit knowledge of the external variable vector, it still can significantly improve the quality of 2D correlation spectra and enables OSC 2D to deal with the problems of losing the quadrature (or out-of-phase) information. In particular, it opens a way to perform QOSC for the spectral dataset without external variables information. The proposed approach should have wide applications in 2D correlation analysis of spectra driven by multiplicative effects in complicated systems in biological, pharmaceutical, and agriculture fields, and so on, where the explicit nature of the external perturbation cannot always be known.

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Benyamine, Charif Abdallah. "On finite sections of the multiplicative Hilbert inequalities." Canadian Mathematical Bulletin, April 8, 2021, 1–9. http://dx.doi.org/10.4153/s0008439521000217.

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Bertram, Alexander, and Martin Grothaus. "Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise." Journal of Evolution Equations 22, no. 1 (March 2022). http://dx.doi.org/10.1007/s00028-022-00773-y.

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AbstractWe provide a complete elaboration of the $$L^2$$ L 2 -Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behavior of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker–Planck framework and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and Röckner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.

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SHEN, MINGMIN, and CHARLES VIAL. "THE MOTIVE OF THE HILBERT CUBE." Forum of Mathematics, Sigma 4 (2016). http://dx.doi.org/10.1017/fms.2016.25.

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The Hilbert scheme $X^{[3]}$ of length-3 subschemes of a smooth projective variety $X$ is known to be smooth and projective. We investigate whether the property of having a multiplicative Chow–Künneth decomposition is stable under taking the Hilbert cube. This is achieved by considering an explicit resolution of the rational map $X^{3}{\dashrightarrow}X^{[3]}$. The case of the Hilbert square was taken care of in Shen and Vial [Mem. Amer. Math. Soc.240(1139) (2016), vii+163 pp]. The archetypical examples of varieties endowed with a multiplicative Chow–Künneth decomposition is given by abelian varieties. Recent work seems to suggest that hyperKähler varieties share the same property. Roughly, if a smooth projective variety $X$ has a multiplicative Chow–Künneth decomposition, then the Chow rings of its powers $X^{n}$ have a filtration, which is the expected Bloch–Beilinson filtration, that is split.

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Journal articles on the topic 'Matrice multiplicative de Hilbert'

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