Relevant bibliographies by topics / Linear transformations

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Linear transformations'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Linear transformations"

1

Michálek, Jiří. "Linear transformations of locally stationary processes." Applications of Mathematics 34, no. 1 (1989): 57–66. http://dx.doi.org/10.21136/am.1989.104334.

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Laffey, Thomas J., and Raphael Loewy. "Linear transformations." Linear and Multilinear Algebra 26, no. 3 (February 1990): 181–86. http://dx.doi.org/10.1080/03081089008817974.

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Pąk, Karol. "Linear Map of Matrices." Formalized Mathematics 16, no. 3 (January 1, 2008): 269–75. http://dx.doi.org/10.2478/v10037-008-0032-0.

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Linear Map of MatricesThe paper is concerned with a generalization of concepts introduced in [13], i.e. introduced are matrices of linear transformations over a finitedimensional vector space. Introduced are linear transformations over a finitedimensional vector space depending on a given matrix of the transformation. Finally, I prove that the rank of linear transformations over a finite-dimensional vector space is the same as the rank of the matrix of that transformation.
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Pąk, Karol. "Linear Transformations of Euclidean Topological Spaces." Formalized Mathematics 19, no. 2 (January 1, 2011): 103–8. http://dx.doi.org/10.2478/v10037-011-0016-3.

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Linear Transformations of Euclidean Topological Spaces We introduce linear transformations of Euclidean topological spaces given by a transformation matrix. Next, we prove selected properties and basic arithmetic operations on these linear transformations. Finally, we show that a linear transformation given by an invertible matrix is a homeomorphism.
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Sullivan, R. P. "Products of nilpotent linear transformations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 124, no. 6 (1994): 1135–50. http://dx.doi.org/10.1017/s0308210500030158.

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In this paper we characterise the linear transformations of an infinite-dimensional vector space that can be written as the product of nilpotent transformations. This and a linear version of Malcev's congruence on transformation semigroups are then used to construct a new class of congruence-free semigroups.
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Fan, Chunpeng, and Jason P. Fine. "Linear Transformation Model With Parametric Covariate Transformations." Journal of the American Statistical Association 108, no. 502 (June 2013): 701–12. http://dx.doi.org/10.1080/01621459.2013.770707.

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WANG, LU, and YIQIANG ZHOU. "‘DECOMPOSING LINEAR TRANSFORMATIONS’." Bulletin of the Australian Mathematical Society 85, no. 1 (November 4, 2011): 172–73. http://dx.doi.org/10.1017/s0004972711002644.

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WANG, LU, and YIQIANG ZHOU. "DECOMPOSING LINEAR TRANSFORMATIONS." Bulletin of the Australian Mathematical Society 83, no. 2 (September 14, 2010): 256–61. http://dx.doi.org/10.1017/s0004972710001711.

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AbstractLet R be the ring of linear transformations of a right vector space over a division ring D. Three results are proved: (1) if |D|>4, then for any a∈R there exists a unit u of R such that a+u,a−u and a−u−1 are units of R; (2) if |D|>3 , then for any a∈R there exists a unit u of R such that both a+u and a−u−1 are units of R; (3) if |D|>2 , then for any a∈R there exists a unit u of R such that both a−u and a−u−1 are units of R. The second result extends the main result in H. Chen, [‘Decompositions of countable linear transformations’, Glasg. Math. J. (2010), doi:10.1017/S0017089510000121] and the third gives an affirmative answer to the question raised in the same paper.
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Friantika, Khasnah Aris, Harina O. L. Monim, and Rium Hilum. "MATRIKS BAKU UNTUK TRANSFORMASI LINIER PADA RUANG VEKTOR DIMENSI TIGA." Jurnal Natural 15, no. 2 (October 1, 2019): 88–93. http://dx.doi.org/10.30862/jn.v15i2.140.

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The linear transformation is a function relating the vector ke . If , then the transformation is called a linear operator. Several examples of linear operators have been introduced since SMA such as reflexive, rotation, compression and expansion and shear. Apart from being introduced in SMA, these linear operators were also introduced to the linear algebra course. Linear transformations studied at the university level include linear transformation in finite dimension vector spaces . The discussion includes how to determine the standard matrix for reflexive linear transformations, rotation, compression and expansion and given shear. Through the column vectors of reflexive, rotation, compression and expansion and shear, a standard matrix of 2x2 size is formed for the corresponding linear transformation. however, in this study, the authors studied linear transformations in dimensioned vector spaces . The results of this study are if known is a vector space with finite and the standard matrix for reflexivity, rotation, expansion, compression and shear is obtained. Each of these linear transformations is performed on x-axis, y-axis and z-axis on to get column vectors. The column vectors as a result of the linear transformation at form the standard matrix for the corresponding linear transformation in the vector space. The standard matrix for linear transformations in the vector space is obtained by determining reflexivity, rotation, expansion, compression and shear. The process of obtaining a standard matrix for linear transformation is carried out by rewriting the standard basis, determining the column vectors, and rearranging them as the standard matrix for each linear transformation in the vector space
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Čadek, Martin. "Form of general pointwise transformations of linear differential equations." Czechoslovak Mathematical Journal 35, no. 4 (1985): 617–24. http://dx.doi.org/10.21136/cmj.1985.102052.

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Dissertations / Theses on the topic "Linear transformations"

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Fretwell, Paul. "Equivalence transformations in linear systems theory." Thesis, Loughborough University, 1986. https://dspace.lboro.ac.uk/2134/33259.

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There is growing interest in infinite frequency structure of linear systems, and transformations preserving this type of structure. Most work has been centred around Generalised State Space (GSS) systems. Two constant equivalence transformations for such systems are Rosenbrock's Restricted System Equivalence (RSE) and Verghese's Strong Equivalence (str.eq.). Both preserve finite and infinite frequency system structure. RSE is over restrictive in that it is constrained to act between systems of the same dimension. While overcoming this basic difficulty str.eq. on the other hand has no closed form description. In this work all these difficulties have been overcome. A constant pencil transformation termed Complete Equivalence (CE) is proposed, this preserves finite elementary divisors and non-unity infinite elementary divisors. Applied to GSS systems CE yields Complete System Equivalence (CSE) which is shown to be a closed form description of str.eq. and is more general than RSE as it relates systems of different dimensions. Equivalence can be described in terms of mappings of the solution sets of the describing differential equations together with mappings of the constrained initial conditions. This provides a conceptually pleasing definition of equivalence. The new equivalence is termed Fundamental Equivalence (FE) and CSE is shown to be a matrix characterisation of it. A polynomial system matrix transformation termed Full Equivalence (fll.e.) is proposed. This relates general matrix polynomials of different dimensions while preserving finite and infinite frequency structure. A definition of infinite zeros is also proposed along with a generalisation of the concept of infinite elementary divisors (IED) from matrix pencils to general polynomial matrices. The IED provide an additional method of dealing with infinite zeros.
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Leggetter, Christopher John. "Improved acoustic modelling for HMMs using linear transformations." Thesis, University of Cambridge, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.361709.

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Manolescu, Crina Iulia. "Lyapunov transformations and control." Thesis, Imperial College London, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.266339.

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Langworthy, Alan Marcel. "A study of some groups of projective transformations." Thesis, University of Oxford, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.365543.

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Kumar, Indra E. "Graphical Applications of Complex and Quaternionic Fractional Linear Transformations." Scholarship @ Claremont, 2016. http://scholarship.claremont.edu/scripps_theses/888.

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The geometric properties of the complex numbers and the quaternions, particularly their behavior under fractional linear transformations, make them very useful for modeling certain types of geometric objects. In this thesis, we will connect the characteristics of fractional linear transformations of both the complex numbers and quaternions to the problem of developing and modifying discrete surfaces for problems in computer graphics and engineering.
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Kinde, Haragewen Abraham. "Contraction and fixed point behavior of certain linear fractional transformations." CSUSB ScholarWorks, 1992. https://scholarworks.lib.csusb.edu/etd-project/1049.

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McInerney, Simon J. "Representations and transformations for multi-dimensional systems." Thesis, Loughborough University, 1999. https://dspace.lboro.ac.uk/2134/28237.

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Multi-dimensional (n-D) systems can be described by matrices whose elements are polynomial in more than one indeterminate. These systems arise in the study of partial differential equations and delay differential equations for example, and have attracted great interest over recent years. Many of the available results have been developed by generalising the corresponding results from the well known 1-D theory. However, this is not always the best approach since there are many differences between 1-D, 2-D and n-D (n > 2) polynomial matrices. This is due mainly to the underlying polynomial ring structure.
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Hansson, Jörgen. "Using Linear Fractional Transformations for Clearance of Flight Control Laws." Thesis, Linköping University, Department of Electrical Engineering, 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-2041.

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Flight Control Systems are often designed in linearization points over a flight envelope and it must be proven to clearance authorities that the system works for different parameter variations and failures all over this envelope.

In this thesis µ-analysis is tried as a complement for linear analysis in the frequency plane. Using this method stability can be guaranteed for all static parameter combinations modelled and linear criteria such as phase and gain margins and most unstable eigenvalue can be included in the analysis. A way of including bounds on the parameter variations using parameter dependent Lyapunov functions is also tried.

To perform µ-analysis the system must be described as a Linear Fractional Transformation (LFT). This is a way of reformulating a parameter dependent system description as an interconnection of a nominal linear time invariant system and a structured parameter block.

A linear and a rational approximation of the system are used to make LFTs. These methods are compared. Four algorithms for calculation of the upper and lower bounds of µ are evaluated. The methods are tried on VEGAS, a SAAB research aircraft model.

µ-analysis works quite well for linear clearance. The rational approximation LFT gives best results and can be cleared for the criteria mentioned above. A combination of the algorithms is used for best results. When the Lyapunov based method is used the size of the problem grows quite fast and, due to numerical problems, stability can only be guaranteed for a reduced model.

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Güldenring, Daniel. "Theory and applications of linear lead transformations in computerised electrocardiology." Thesis, University of Ulster, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.603537.

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An electrocardiogram (ECG) can be used for the identification of different heart diseases. Linear lead transformations (LLTs) allow for the estimation of unrecorded ECG leads through linear combinations of a number of recorded ECG leads. This estimation relies upon statistical inter-lead correlations. Reduced lead systems (RLSs) commonly utilise LLTs for the estimation of the full 12-lead ECG from a reduced number of recorded ECG leads. The RLS used within this thesis utilised the recorded basis leads I, II, V2 and V5 for the estimation of the unrecorded target leads VI, V3, V 4 and V6. An assessment of whether the utilisation of statistical short-term autocorrelations, in addition to the commonly used inter-lead correlations, can improve the estimation performance (EP) of the above RLS has been conducted. The utilisation of statistical short-term autocorrelations was found not to improve the EP during episodes of acute myocardial ischemia (AMI). The effect of AMI on the EP of the aforementioned RLS has been investigated. It was found that AMI can reduce the EP. The inability of LLTs to fully describe the electrical volume conductor properties of the human trunk was found to be the likely source for the reduction in the EP. The EP achieved by the above RLS, has been assessed during episodes of AMI. It was found that a similar EP for target leads VI and V6 was achieved regardless of whether patient-specific or generalised transformation weights were employed. An LLT that allows for the estimation of the Frank vectorcardiogram (yCG) from the Mason-Likar (ML) 12-1ead ECG was developed. The developed LL T and the Kors matrix were used for the estimation of the Frank VCG from ML 12-lead ECG data. The developed LLT was found to allow the determination of the VCG parameters 'spatial ventricular gradient' and 'spatial QRS-T angle' with higher accuracy than when using the Kors matrix.
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Guldenring, Daniel. "Theory and applications of linear lead transformations in computerised electrocardiography." Thesis, Ulster University, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.665848.

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Books on the topic "Linear transformations"

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Matrices and linear transformations. 2nd ed. New York: Dover, 1990.

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Farenick, Douglas R. Algebras of Linear Transformations. New York, NY: Springer New York, 2001. http://dx.doi.org/10.1007/978-1-4613-0097-7.

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Campbell, S. L. Generalized inverses of linear transformations. New York: Dover Publications, 1991.

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Campbell, S. L. Generalized inverses of linear transformations. Philadelphia: Society for Industrial and Applied Mathematics, 2009.

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Amir-Moéz, A. R. Extreme properties of linear transformations. Washington, NJ: Polygonal Pub., 1990.

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Linear algebra and projective geometry. Mineola, N.Y: Dover Publications, 2005.

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Nillsen, Rodney. Difference spacesand invariant linear forms. Berlin: Springer-Verlag, 1994.

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Global properties of linear ordinary differential equations. Dordrecht: Kluwer Academic Publishers, 1991.

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Jerri, Abdul J. Linear difference equations with discrete transform methods. Dordrecht: Kluwer Academic, 1996.

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Loon, P. M. van. Continuous decoupling transformations for linear boundary value problems. [Amsterdam, the Netherlands]: Centrum voor Wiskunde en Informatica, 1988.

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Book chapters on the topic "Linear transformations"

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Jeff, Suzuki. "Transformations." In Linear Algebra, 81–114. First edition. | Boca Raton, FL : CRC Press, 2021. | Series: Textbooks in mathematics: CRC Press, 2021. http://dx.doi.org/10.1201/9780429284984-3.

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Kwak, Jin Ho, and Sungpyo Hong. "Linear Transformations." In Linear Algebra, 117–56. Boston, MA: Birkhäuser Boston, 2004. http://dx.doi.org/10.1007/978-0-8176-8194-4_4.

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Kwak, Jin Ho, and Sungpyo Hong. "Linear Transformations." In Linear Algebra, 121–59. Boston, MA: Birkhäuser Boston, 1997. http://dx.doi.org/10.1007/978-1-4757-1200-1_4.

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Nair, M. Thamban, and Arindama Singh. "Linear Transformations." In Linear Algebra, 51–105. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-13-0926-7_2.

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Smith, Larry. "Linear Transformations." In Linear Algebra, 85–112. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-1670-4_8.

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Roman, Steven. "Linear Transformations." In Advanced Linear Algebra, 59–85. New York, NY: Springer New York, 2008. http://dx.doi.org/10.1007/978-0-387-72831-5_2.

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Roman, Steven. "Linear Transformations." In Advanced Linear Algebra, 45–62. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4757-2178-2_3.

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Bressoud, David M. "Linear Transformations." In Second Year Calculus, 139–70. New York, NY: Springer New York, 1991. http://dx.doi.org/10.1007/978-1-4612-0959-1_6.

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Anastassiou, George A., and Iuliana F. Iatan. "Linear Transformations." In Intelligent Systems Reference Library, 91–134. Cham: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-01967-3_3.

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Towers, David A. "Linear Transformations." In Guide to Linear Algebra, 124–59. London: Macmillan Education UK, 1988. http://dx.doi.org/10.1007/978-1-349-09318-2_6.

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Conference papers on the topic "Linear transformations"

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Alexa, Marc. "Linear combination of transformations." In the 29th annual conference. New York, New York, USA: ACM Press, 2002. http://dx.doi.org/10.1145/566570.566592.

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Coggins, J. M. "Non-linear feature space transformations." In IEE Colloquium on Applied Statistical Pattern Recognition. IEE, 1999. http://dx.doi.org/10.1049/ic:19990374.

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Swain, J. J. "Augmenting linear control variates using transformations." In the 21st conference. New York, New York, USA: ACM Press, 1989. http://dx.doi.org/10.1145/76738.76796.

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Kim, T. K., J. Kittler, H. C. Kim, and S. C. Kee. "Discriminant Analysis by Locally Linear Transformations." In British Machine Vision Conference 2003. British Machine Vision Association, 2003. http://dx.doi.org/10.5244/c.17.13.

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Ying, Leslie, and Yi Ming Zou. "Linear transformations and Restricted Isometry Property." In ICASSP 2009 - 2009 IEEE International Conference on Acoustics, Speech and Signal Processing. IEEE, 2009. http://dx.doi.org/10.1109/icassp.2009.4960245.

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Tomczak, Jakub M., Ewelina Węglarz-Tomczak, and Agoston E. Eiben. "Differential Evolution with Reversible Linear Transformations." In GECCO '20: Genetic and Evolutionary Computation Conference. New York, NY, USA: ACM, 2020. http://dx.doi.org/10.1145/3377929.3389972.

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Li, Chao, Wei He, Longhao Yuan, Zhun Sun, and Qibin Zhao. "Guaranteed Matrix Completion Under Multiple Linear Transformations." In 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, 2019. http://dx.doi.org/10.1109/cvpr.2019.01139.

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Koren, Y., and L. Carmel. "Visualization of labeled data using linear transformations." In IEEE Symposium on Information Visualization 2003. IEEE, 2003. http://dx.doi.org/10.1109/infvis.2003.1249017.

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Wang, Xiaolin, Guoqin Chen, and Jianqin Zhou. "A Note on Linear Transformations in Cryptography." In 2009 International Symposium on Information Engineering and Electronic Commerce (IEEC). IEEE, 2009. http://dx.doi.org/10.1109/ieec.2009.41.

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Wolf, Kurt Bernardo, Piotr Kielanowski, Anatol Odzijewicz, Martin Schlichenmaier, and Theodore Voronov. "Finite Hamiltonian Systems: Linear Transformations and Aberrations." In GEOMETRIC METHODS IN PHYSICS. AIP, 2008. http://dx.doi.org/10.1063/1.3043865.

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Reports on the topic "Linear transformations"

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Renault, Marc. Linear Transformations of Points. Washington, DC: The MAA Mathematical Sciences Digital Library, April 2009. http://dx.doi.org/10.4169/loci003299.

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Rao, C. R. Linear Transformations, Projection Operators and Generalized Inverses; A Geometric Approach. Fort Belvoir, VA: Defense Technical Information Center, March 1988. http://dx.doi.org/10.21236/ada197608.

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Brigola, R. Stochastic Filtering Solutions for Ill-Posed Linear Problems and Their Extension to Measurable Transformations. Fort Belvoir, VA: Defense Technical Information Center, March 1987. http://dx.doi.org/10.21236/ada186016.

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Leger, J. R., and W. C. Goltsos. Geometrical Transformation of Linear Diode-Laser Arrays for Longitudinal Pumping of Solid-State Lasers. Fort Belvoir, VA: Defense Technical Information Center, May 1992. http://dx.doi.org/10.21236/ada253415.

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Tsoupas, Nicholaos, Francis Meot, and Haixin Huang. Transformation of Spinors in Accelerators and Beam Transfer Lines. Office of Scientific and Technical Information (OSTI), December 2020. http://dx.doi.org/10.2172/1749905.

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Viall, Kenneth E. Big Blue Arrows: Lines of Information and the Transformation Force. Fort Belvoir, VA: Defense Technical Information Center, February 2001. http://dx.doi.org/10.21236/ada392266.

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Angulo Rodríguez, Emilio, and Ariel Yépez-García. The Role of Natural Gas in Energy Transition. Inter-American Development Bank, November 2020. http://dx.doi.org/10.18235/0002868.

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As of 2004 and continuously to this day, the annual growth rate of renewable sources has been greater than that of all fossil fuels combined. In the midst of this transition to cleaner energy, natural gas is the only fossil fuel that has increased its share in the global energy matrix. Technological changes in the LNG supply chain, as well as transformations in the global natural gas market, largely explain this growth. This publication provides evidence on the fundamental role that natural gas plays in the energy transition, given that: (i) its greenhouse gas emissions are substantially lower than those of oil and coal; (ii) it provides the firm power necessary to complement intermittent renewable energies; (iii) it is particularly safe compared to other fossil fuels. In line with these attributes, the International Energy Agency projects that the share of natural gas in the global energy matrix by 2040 will remain stable (around 24%), even in its Sustainable Development Scenario, which would allow to meet the goals established in the Paris Agreement.
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Saville, Alan, and Caroline Wickham-Jones, eds. Palaeolithic and Mesolithic Scotland : Scottish Archaeological Research Framework Panel Report. Society for Antiquaries of Scotland, June 2012. http://dx.doi.org/10.9750/scarf.06.2012.163.

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Why research Palaeolithic and Mesolithic Scotland? Palaeolithic and Mesolithic archaeology sheds light on the first colonisation and subsequent early inhabitation of Scotland. It is a growing and exciting field where increasing Scottish evidence has been given wider significance in the context of European prehistory. It extends over a long period, which saw great changes, including substantial environmental transformations, and the impact of, and societal response to, climate change. The period as a whole provides the foundation for the human occupation of Scotland and is crucial for understanding prehistoric society, both for Scotland and across North-West Europe. Within the Palaeolithic and Mesolithic periods there are considerable opportunities for pioneering research. Individual projects can still have a substantial impact and there remain opportunities for pioneering discoveries including cemeteries, domestic and other structures, stratified sites, and for exploring the huge evidential potential of water-logged and underwater sites. Palaeolithic and Mesolithic archaeology also stimulates and draws upon exciting multi-disciplinary collaborations. Panel Task and Remit The panel remit was to review critically the current state of knowledge and consider promising areas of future research into the earliest prehistory of Scotland. This was undertaken with a view to improved understanding of all aspects of the colonization and inhabitation of the country by peoples practising a wholly hunter-fisher-gatherer way of life prior to the advent of farming. In so doing, it was recognised as particularly important that both environmental data (including vegetation, fauna, sea level, and landscape work) and cultural change during this period be evaluated. The resultant report, outlines the different areas of research in which archaeologists interested in early prehistory work, and highlights the research topics to which they aspire. The report is structured by theme: history of investigation; reconstruction of the environment; the nature of the archaeological record; methodologies for recreating the past; and finally, the lifestyles of past people – the latter representing both a statement of current knowledge and the ultimate aim for archaeologists; the goal of all the former sections. The document is reinforced by material on-line which provides further detail and resources. The Palaeolithic and Mesolithic panel report of ScARF is intended as a resource to be utilised, built upon, and kept updated, hopefully by those it has helped inspire and inform as well as those who follow in their footsteps. Future Research The main recommendations of the panel report can be summarized under four key headings:  Visibility: Due to the considerable length of time over which sites were formed, and the predominant mobility of the population, early prehistoric remains are to be found right across the landscape, although they often survive as ephemeral traces and in low densities. Therefore, all archaeological work should take into account the expectation of Palaeolithic and Mesolithic ScARF Panel Report iv encountering early prehistoric remains. This applies equally to both commercial and research archaeology, and to amateur activity which often makes the initial discovery. This should not be seen as an obstacle, but as a benefit, and not finding such remains should be cause for question. There is no doubt that important evidence of these periods remains unrecognised in private, public, and commercial collections and there is a strong need for backlog evaluation, proper curation and analysis. The inadequate representation of Palaeolithic and Mesolithic information in existing national and local databases must be addressed.  Collaboration: Multi-disciplinary, collaborative, and cross- sector approaches must be encouraged – site prospection, prediction, recognition, and contextualisation are key areas to this end. Reconstructing past environments and their chronological frameworks, and exploring submerged and buried landscapes offer existing examples of fruitful, cross-disciplinary work. Palaeolithic and Mesolithic archaeology has an important place within Quaternary science and the potential for deeply buried remains means that geoarchaeology should have a prominent role.  Innovation: Research-led projects are currently making a substantial impact across all aspects of Palaeolithic and Mesolithic archaeology; a funding policy that acknowledges risk and promotes the innovation that these periods demand should be encouraged. The exploration of lesser known areas, work on different types of site, new approaches to artefacts, and the application of novel methodologies should all be promoted when engaging with the challenges of early prehistory.  Tackling the ‘big questions’: Archaeologists should engage with the big questions of earliest prehistory in Scotland, including the colonisation of new land, how lifestyles in past societies were organized, the effects of and the responses to environmental change, and the transitions to new modes of life. This should be done through a holistic view of the available data, encompassing all the complexities of interpretation and developing competing and testable models. Scottish data can be used to address many of the currently topical research topics in archaeology, and will provide a springboard to a better understanding of early prehistoric life in Scotland and beyond.
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