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Melkemi, Lamine, Abdallah Benaissa, and Rachid Benacer. "Factorisation QR des matrices de Tchebychev–Vandermonde confluentes." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 330, no. 2 (January 2000): 147–52. http://dx.doi.org/10.1016/s0764-4442(00)00144-0.
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Jacobson, Bernard, and Robert J. Wisner. "Matrix number theory. I. Factorisation of $2\times 2$ unimodular matrices." Publicationes Mathematicae Debrecen 13, no. 1-4 (July 1, 2022): 67–72. http://dx.doi.org/10.5486/pmd.1966.13.1-4.08.
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Liew, How Hui, Wei Shean Ng, and Huey Voon Chen. "Investigating the feature extraction capabilities of non-negative matrix factorisation algorithms for black-and-white images." ITM Web of Conferences 67 (2024): 01031. http://dx.doi.org/10.1051/itmconf/20246701031.
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Nonnegative matrix factorisation (NMF) is a class of matrix factorisation methods to approximate a nonnegative matrix as a product of two nonnegative matrices. To derive NMF algorithms, the optimisation problems for NMF are developed and the divergence used in the optimisation problems can have many forms. The β-divergence is the most popular and is used in this research. The NMF algorithms derived from the β-divergence have a few hyperparameters including the rank and the initial conditions. This paper surveyed on the software implementations of the NMF algorithms and then applied the open source software implementations of Frobenius norm based NMF algorithm, KL divergence based NMF algorithm and binary matrix factorisation (BMF) with fixed ranks to three classes of black-and-white images. For black-and-white images with a lot of common features (like MNIST), KL divergence NMF with appropriate initial guess is empirically found to be best NMF algorithm for black-and-white image feature extraction compare to other NMF algorithms. All NMF algorithms for data with little to no common features are useful in generating feature images which can be used to inspire art design as well as in the realm of computer vision.
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Bakonyi, Mihály. "On Gaussian elimination and determinant formulas for matrices with chordal inverses." Bulletin of the Australian Mathematical Society 46, no. 3 (December 1992): 435–40. http://dx.doi.org/10.1017/s0004972700012090.
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In this paper a formula is obtained for the entries of the diagonal factor in the U D L factorisation of an invertible operator matrix in the case when its inverse has a chordal graph. As a consequence, in the finite dimensional case a determinant formula is obtained in terms of some key principal minors. After a cancellation process this formula leads to a determinant formula from an earlier paper by W.W. Barrett and C.R. Johnson, deriving in this way a different and shorter proof of their result. Finally, an algorithmic method of constructing minimal vertex separators of chordal graphs is presented.
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Carbonell-Caballero, José, Antonio López-Quílez, David Conesa, and Joaquín Dopazo. "Deciphering Genomic Heterogeneity and the Internal Composition of Tumour Activities through a Hierarchical Factorisation Model." Mathematics 9, no. 21 (November 8, 2021): 2833. http://dx.doi.org/10.3390/math9212833.
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Genomic heterogeneity constitutes one of the most distinctive features of cancer diseases, limiting the efficacy and availability of medical treatments. Tumorigenesis emerges as a strongly stochastic process, producing a variable landscape of genomic configurations. In this context, matrix factorisation techniques represent a suitable approach for modelling such complex patterns of variability. In this work, we present a hierarchical factorisation model conceived from a systems biology point of view. The model integrates the topology of molecular pathways, allowing to simultaneously factorise genes and pathways activity matrices. The protocol was evaluated by using simulations, showing a high degree of accuracy. Furthermore, the analysis with a real cohort of breast cancer patients depicted the internal composition of some of the most relevant altered biological processes in the disease, describing gene and pathway level strategies and their observed combinations in the population of patients. We envision that this kind of approaches will be essential to better understand the hallmarks of cancer.
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Spencer, N. M., and V. V. Anh. "Spectral factorisation and prediction of multivariate processes with time-dependent rational spectral density matrices." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 33, no. 2 (October 1991): 192–210. http://dx.doi.org/10.1017/s0334270000006998.
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AbstractThis paper considers discrete multivariate processes with time-dependent rational spectral density matrices and gives a solution to the spectral factorisation problem. As a result, the corresponding state space representation for the process is obtained. The relationship between multivariate processes with time-dependent rational spectral density matrix functions and multivariate ARMA processes with time-dependent coefficients is discussed. Solutions for the prediction problem are given for the case when only finite data is available and the case when the whole history of the process is known.
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Hamel, Angèle M., and Ronald C. King. "Half-turn symmetric alternating sign matrices and Tokuyama type factorisation for orthogonal group characters." Journal of Combinatorial Theory, Series A 131 (April 2015): 1–31. http://dx.doi.org/10.1016/j.jcta.2014.11.005.
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Prößdorf, S., and F. O. Speck. "A factorisation procedure for two by two matrix functions on the circle with two rationally independent entries." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 115, no. 1-2 (1990): 119–38. http://dx.doi.org/10.1017/s0308210500024616.
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SynopsisThe aim of this paper is the explicit canonical or standard factorisation of matrix functions with Wiener algebra elements. The present approach covers all regular 2 × 2 matrices where two entries are arbitrary and the remaining two are linear combinations of the former with rational coefficient functions. It is based on the knowledge of how to factorise scalar functions and rational matrix functions. In general, one also needs the approximation of any scalar Wiener algebra function with a rational function. However, this can be easily circumvented in many applications by intuitive manipulations with rational matrix functions.
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Leipnik, R. B. "Reduction of second order linear dynamical systems, with large dissipation, by state variable transformations." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 32, no. 2 (October 1990): 207–22. http://dx.doi.org/10.1017/s0334270000008432.
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AbstractLinear dynamical systems of the Rayleigh form are transformed by linear state variable transformations , where A and B are chosen to simplify analysis and reduce computing time. In particular, A is essentially a square root of M, and B is a Lyapunov quotient of C by A. Neither K nor C is required to be symmetric, nor is C small. The resulting state-space systems are analysed by factorisation of the evolution matrices into reducible factors. Eigenvectors and eigenvalues are determined by these factors. Conditions for further simplification are derived in terms of Kronecker determinants. These results are compared with classical reductions of Rayleigh, Duncan, and Caughey, which are reviewed at the outset.
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Abou-Zleikha, Mohamed, and Noor Shaker. "PaTux: An Authoring Tool for Level Design through Pattern Customisation Using Non-Negative Matrix Factorization." Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment 10, no. 1 (June 29, 2021): 103–4. http://dx.doi.org/10.1609/aiide.v10i1.12726.
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We present a demonstration of PaTux, an authoring tool for designing levels in SuperTux game through combining patterns. PaTux allows game designers to specify the design of their levels using patterns extracted from training level samples. The Non-negative Matrix Factorisation (NMF) method is utilised to approximate pattern and weight matrices from the training data. The patterns are visualised for designers to choose from and the changes made on the level structure are visualised in realtime. The designer can also specify the weight of each pattern permitting exploration of a wider variety. The data used to train the model can also be specified by the designer resulting in learning a new set of patterns. The system also suggests variations for a given design. When the designer is satisfied with the design, the system allows loading the resultant level in the game to be played.
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Dinkelbach, Jan, Lennart Schumacher, Lukas Razik, Andrea Benigni, and Antonello Monti. "Factorisation Path Based Refactorisation for High-Performance LU Decomposition in Real-Time Power System Simulation." Energies 14, no. 23 (November 30, 2021): 7989. http://dx.doi.org/10.3390/en14237989.
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The integration of renewable energy sources into modern power systems requires simulations with smaller step sizes, larger network models and the incorporation of complex nonlinear component models. These features make it more difficult to meet computation time requirements in real-time simulations and have motivated the development of high-performance LU decomposition methods. Since nonlinear component models cause numerical variations in the system matrix between simulation steps, this paper places a particular focus on the recomputation of LU decomposition, i.e., on the refactorisation step. The main contribution is the adoption of a factorisation path algorithm for partial refactorisation, which takes into account that only a subset of matrix entries change their values. The approach is integrated into the modern LU decomposition method NICSLU and benchmarked against the methods SuperLU and KLU. A performance analysis was carried out considering benchmark as well as real power systems. The results show the significant speedup of refactorisation computation times in use cases involving system matrices of different sizes, a variety of sparsity patterns and different ratios of numerically varying matrix entries. Consequently, the presented high-performance LU decomposition method can assist in meeting computation time requirements in real-time simulations of modern power systems.
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Farenick, Douglas, and Michelle McBurney. "Toeplitz separability, entanglement, and complete positivity using operator system duality." Proceedings of the American Mathematical Society, Series B 10, no. 10 (April 10, 2023): 114–28. http://dx.doi.org/10.1090/bproc/163.
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A new proof is presented of a theorem of L. Gurvits [LANL Unclassified Technical Report (2001), LAUR–01–2030], which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. The proof of the Gurvits separation theorem is achieved by making use of the structure of the operator system dual of the operator system C ( S 1 ) ( n ) C(S^1)^{(n)} of n × n n\times n Toeplitz matrices over the complex field, and by determining precisely the structure of the generators of the extremal rays of the positive cones of the operator systems C ( S 1 ) ( n ) ⊗ min B ( H ) C(S^1)^{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}) and C ( S 1 ) ( n ) ⊗ min B ( H ) C(S^1)_{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}) , where H \mathcal {H} is an arbitrary Hilbert space and C ( S 1 ) ( n ) C(S^1)_{(n)} is the operator system dual of C ( S 1 ) ( n ) C(S^1)^{(n)} . Our approach also has the advantage of providing some new information concerning positive Toeplitz matrices whose entries are from B ( H ) \mathcal {B}(\mathcal {H}) when H \mathcal {H} has infinite dimension. In particular, we prove that normal positive linear maps ψ \psi on B ( H ) \mathcal {B}(\mathcal {H}) are partially completely positive in the sense that ψ ( n ) ( x ) \psi ^{(n)}(x) is positive whenever x x is a positive n × n n\times n Toeplitz matrix with entries from B ( H ) \mathcal {B}(\mathcal {H}) . We also establish a certain factorisation theorem for positive Toeplitz matrices (of operators), showing an equivalence between the Gurvits approach to separation and an earlier approach of T. Ando [Acta Sci. Math. (Szeged) 31 (1970), pp. 319–334] to universality.
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Vibitha Kochamani, V., and P. L. Lilly. "Modified Form of Cayley Hash Function." Asian Journal of Engineering and Applied Technology 8, no. 2 (May 5, 2019): 34–36. http://dx.doi.org/10.51983/ajeat-2019.8.2.1142.
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In 1994 (AT CRYPTO94) introduced the celebrated Zemor-Tillich hash function over SL2(F2n) is mathematically very efficient and simple method but now finally it was broken by Grassl et al., 2011. Yet with a new choice of generators Zemor-Tillich constructions still remains of interest and a lot of construction was based on this type of hash function was created. One of our new construction is the devised Hash Function as follows: to an arbitrary text of {0, 1} *, associate the string of {A, B} obtained by substituting 0 for A and 1 for B, then assign to A and B values of adequately chosen matrices of Heis(Z). Now, in this paper we suggest a new version of a Cayley hash function using a discrete Heisenberg group. The Hashed value is the computed product. We improved the security Properties of the Cayley Hash Function. Here we hold a different concept to form a Factorisation Problem harder. We hold an efficient way to impose limits on the type of factorisations for attacking H.
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Wang, Wei, and Elena Kozlova. "Equilibration of athletes’ body state in the course of increasing the training intensity." SPORT TK-Revista EuroAmericana de Ciencias del Deporte 13 (April 17, 2024): 28. http://dx.doi.org/10.6018/sportk.555871.
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Recovery of an athlete after injury allows to achieve either the previous results or relatively equal. Moreover, the prescribed drugs or recovery exercises are offered in courses without the possibility of process adjustments. The relevance of the study is determined primarily by a large number of universal methods of restoration, if necessary, development of more patient-specific approach. The novelty of the study is determined by the fact that the human body is a dynamic system, the state of which is different at any time. The authors developed the idea of a stochastic process problem in the form of a system with a variable state, which can be utilized to generate a process, in order to build a technique for recognizing and predicting the state of dynamic systems. The authors showed that the dynamic system of the body can be considered not only as a process element, but also as a structure that allows to fully achieve recovery during the training period of athletes and their off-season indicators. The paper has developed a model that allows to predict the state of the dynamic system and harmonise the load and the recovery process of the athlete's body. The developed method for identifying and predicting the state of dynamical systems with the improvement of the method of covariance functions factorisation makes it possible to determine the main matrices of a dynamical system using the Riccati equation.
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Amil, N., M. T. Latif, M. F. Khan, and M. Mohamad. "Meteorological-gaseous influences on seasonal PM<sub>2.5</sub> variability in the Klang Valley urban-industrial environment." Atmospheric Chemistry and Physics Discussions 15, no. 18 (September 30, 2015): 26423–79. http://dx.doi.org/10.5194/acpd-15-26423-2015.
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Abstract. This study attempts to investigate the fine particulate matter (PM2.5) variability in the Klang Valley urban-industrial environment. In total, 94 daily PM2.5 samples were collected during a one-year campaign from August 2011 to July 2012, covering all four seasons. The samples were analysed for various inorganic components and black carbon. The chemical compositions were statistically analysed and the aerosol pattern was characterised using descriptive analysis, correlation matrices, enrichment factors (EF), stoichiometric analysis and chemical mass closure (CMC). For source apportionment purposes, a combination of positive matrix factorisation (PMF) and multi-linear regression (MLR) was employed. Further, meteorological-gaseous parameters were incorporated into each analysis for improved assessment. The results showed that PM2.5 mass averaged at 28 ± 18 μg m−3, 2.8 fold higher than the World Health Organisation (WHO) annual guideline. On a daily basis, the PM2.5 mass ranged between 6 and 118 μg m−3 with 43 % exceedance of the daily WHO guideline. The North-East monsoon (NE) was the only season with < 50 % sample exceedance of the daily WHO guideline. On an annual scale, PM2.5 mass correlated positively with temperature (T) and wind speed (WS) but negatively with relative humidity (RH). With the exception of NOx, the gases analysed (CO, NO2, NO and SO2) were found to significantly influence the PM2.5 mass. Seasonal variability unexpectedly showed that rainfall, WS and wind direction (WD) did not significantly correlate with PM2.5 mass. Further analysis on the PM2.5 / PM10, PM2.5 / TSP and PM10 / TSP ratios reveal that meteorological parameters only greatly influenced the coarse particles (PM > 2.5μm) and less so the fine particles at the site. Chemical composition showed that both primary and secondary pollutants of PM2.5 are equally important, albeit with seasonal variability. The CMC components identified were: black carbon (BC) > secondary inorganic aerosols (SIA) > dust > trace elements (TE) > sea salt > K+. The EF analysis distinguished two groups of trace elements: those with anthropogenic sources (Pb, Se, Zn, Cd, As, Bi, Ba, Cu, Rb, V and Ni) and those with a crustal source (Sr, Mn, Co and Li). The five identified factors resulting from PMF 5.0 were: (1) combustion of engine oil; (2) mineral dust; (3) mixed SIA and biomass burning; (4) mixed traffic and industrial; and (5) sea salt. Each of these sources had an annual mean contribution of 17, 14, 42, 10 and 17 %, respectively. The dominance of each identified source largely varied with changing season and a few factors were in agreement with the CMC, EF and stoichiometric analysis, accordingly. In relation to meteorological-gaseous parameters, PM2.5 sources were influenced by different parameters during different seasons. In addition, two air pollution episodes (HAZE) revealed the influence of local and/or regional sources. Overall, our study clearly suggests that the chemical constituents and sources of PM2.5 were greatly influenced and characterised by meteorological and gaseous parameters which largely vary with season.
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Amil, Norhaniza, Mohd Talib Latif, Md Firoz Khan, and Maznorizan Mohamad. "Seasonal variability of PM<sub>2.5</sub> composition and sources in the Klang Valley urban-industrial environment." Atmospheric Chemistry and Physics 16, no. 8 (April 29, 2016): 5357–81. http://dx.doi.org/10.5194/acp-16-5357-2016.
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Abstract. This study investigates the fine particulate matter (PM2.5) variability in the Klang Valley urban-industrial environment. In total, 94 daily PM2.5 samples were collected during a 1-year campaign from August 2011 to July 2012. This is the first paper on PM2.5 mass, chemical composition and sources in the tropical environment of Southeast Asia, covering all four seasons (distinguished by the wind flow patterns) including haze events. The samples were analysed for various inorganic components and black carbon (BC). The chemical compositions were statistically analysed and the temporal aerosol pattern (seasonal) was characterised using descriptive analysis, correlation matrices, enrichment factor (EF), stoichiometric analysis and chemical mass closure (CMC). For source apportionment purposes, a combination of positive matrix factorisation (PMF) and multi-linear regression (MLR) was employed. Further, meteorological–gaseous parameters were incorporated into each analysis for improved assessment. In addition, secondary data of total suspended particulate (TSP) and coarse particulate matter (PM10) sampled at the same location and time with this study (collected by Malaysian Meteorological Department) were used for PM ratio assessment. The results showed that PM2.5 mass averaged at 28 ± 18 µg m−3, 2.8-fold higher than the World Health Organisation (WHO) annual guideline. On a daily basis, the PM2.5 mass ranged between 6 and 118 µg m−3 with the daily WHO guideline exceeded 43 % of the time. The north-east (NE) monsoon was the only season with less than 50 % sample exceedance of the daily WHO guideline. On an annual scale, PM2.5 mass correlated positively with temperature (T) and wind speed (WS) but negatively with relative humidity (RH). With the exception of NOx, the gases analysed (CO, NO2, NO and SO2) were found to significantly influence the PM2.5 mass. Seasonal variability unexpectedly showed that rainfall, WS and wind direction (WD) did not significantly correlate with PM2.5 mass. Further analysis on the PM2.5 ∕ PM10, PM2.5 ∕ TSP and PM10 ∕ TSP ratios reveal that meteorological parameters only greatly influenced the coarse particles (particles with an aerodynamic diameter of greater than 2.5 µm) and less so the fine particles at the site. Chemical composition showed that both primary and secondary pollutants of PM2.5 are equally important, albeit with seasonal variability. The CMC components identified were in the decreasing order of (mass contribution) BC > secondary inorganic aerosols (SIA) > dust > trace elements > sea salt > K+. The EF analysis distinguished two groups of trace elements: those with anthropogenic sources (Pb, Se, Zn, Cd, As, Bi, Ba, Cu, Rb, V and Ni) and those with a crustal source (Sr, Mn, Co and Li). The five identified factors resulting from PMF 5.0 were (1) combustion of engine oil, (2) mineral dust, (3) mixed SIA and biomass burning, (4) mixed traffic and industrial and (5) sea salt. Each of these sources had an annual mean contribution of 17, 14, 42, 10 and 17 % respectively. The dominance of each identified source largely varied with changing season and a few factors were in agreement with the CMC, EF and stoichiometric analysis, accordingly. In relation to meteorological–gaseous parameters, PM2.5 sources were influenced by different parameters during different seasons. In addition, two air pollution episodes (HAZE) revealed the influence of local and/or regional sources. Overall, our study clearly suggests that the chemical constituents and sources of PM2.5 were greatly influenced and characterised by meteorological and gaseous parameters which vary greatly with season.
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Câmara, M. Cristina, and Gabriel Lopes Cardoso. "Riemann-Hilbert problems, Toeplitz operators and ergosurfaces." Journal of High Energy Physics 2024, no. 6 (June 5, 2024). http://dx.doi.org/10.1007/jhep06(2024)027.
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Abstract The Riemann-Hilbert approach, in conjunction with the canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices, enables one to obtain explicit solutions to the non-linear field equations of some gravitational theories. These solutions are encoded in the elements of a matrix M depending on the Weyl coordinates ρ and v, determined by that factorisation. We address here, for the first time, the underlying question of what happens when a canonical Wiener-Hopf factorisation does not exist, using the close connection of Wiener-Hopf factorisation with Toeplitz operators to study this question. For the case of rational monodromy matrices, we prove that the non-existence of a canonical Wiener-Hopf factorisation determines curves in the (ρ, v) plane on which some elements of M(ρ, v) tend to infinity, but where the space-time metric may still be well behaved. In the case of uncharged rotating black holes in four space-time dimensions and, for certain choices of coordinates, in five space-time dimensions, we show that these curves correspond to their ergosurfaces.
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Dobson, Liam, and Stefan Kolb. "Factorisation of quasi K-matrices for quantum symmetric pairs." Selecta Mathematica 25, no. 4 (October 2019). http://dx.doi.org/10.1007/s00029-019-0508-5.
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Abstract The theory of quantum symmetric pairs provides a universal K-matrix which is an analog of the universal R-matrix for quantum groups. The main ingredient in the construction of the universal K-matrix is a quasi K-matrix which has so far only been constructed recursively. In this paper we restrict to the cases where the underlying Lie algebra is $$\mathfrak {sl}_n$$ sl n or the Satake diagram has no black dots. In these cases we give an explicit formula for the quasi K-matrix as a product of quasi K-matrices for Satake diagrams of rank one. This factorisation depends on the restricted Weyl group of the underlying symmetric Lie algebra in the same way as the factorisation of the quasi R-matrix depends on the Weyl group of the Lie algebra. We conjecture that our formula holds in general.
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Teal, Paul D. "Matrix factorisations for the estimation of NMR relaxation distributions." Diffusion Fundamentals 22 (December 31, 2014). http://dx.doi.org/10.62721/diffusion-fundamentals.22.837.
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The two most successful methods of estimating the distribution of NMR relaxation times from two dimensional data are firstly a data compression stage followed by application of the Butler-Reeds-Dawson (BRD) algorithm, and secondly a primal dual interior point method using a preconditioned conjugate gradient (PCG). Both of these methods have been presented in the literature as requiring a truncated singular value decomposition of matrices representing the exponential kernels. Other matrix factorisations are applicable to each of these algorithms, and which demonstrate the different fundamental principles behind the operation of the algorithms. In the case of the data compression approach the most appropriate matrix decomposition specifically designed for this task is the rank-revealing QR (RRQR) factorisation. In the case of the interior point method, the most appropriate method is the LDL factorisation with diagonal pivoting, also known as the Bunch-Kaufman-Parlett factorisation. The details of these differences are discussed, and the performances of the algorithms are compared numerically.
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Shams, Sulthana, and Douglas Leith. "Evaluating Impact of User-Cluster Targeted Attacks in Matrix Factorisation Recommenders." ACM Transactions on Recommender Systems, June 21, 2024. http://dx.doi.org/10.1145/3674157.
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In practice, users of a Recommender System (RS) fall into a few clusters based on their preferences. In this work, we conduct a systematic study on user-cluster targeted data poisoning attacks on Matrix Factorisation (MF) based RS, where an adversary injects fake users with falsely crafted user-item feedback to promote an item to a specific user cluster. We analyse how user and item feature matrices change after data poisoning attacks and identify the factors that influence the effectiveness of the attack on these feature matrices. We demonstrate that the adversary can easily target specific user clusters with minimal effort and that some items are more susceptible to attacks than others. Our theoretical analysis has been validated by the experimental results obtained from two real-world datasets. Our observations from the study could serve as a motivating point to design a more robust RS.
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Matsumoto, Sho, and Jonathan Novak. "Unitary Matrix Integrals, Primitive Factorizations, and Jucys-Murphy Elements." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AN,..., Proceedings (January 1, 2010). http://dx.doi.org/10.46298/dmtcs.2879.
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International audience A factorization of a permutation into transpositions is called "primitive'' if its factors are weakly ordered.We discuss the problem of enumerating primitive factorizations of permutations, and its place in the hierarchy of previously studied factorization problems. Several formulas enumerating minimal primitive and possibly non-minimal primitive factorizations are presented, and interesting connections with Jucys-Murphy elements, symmetric group characters, and matrix models are described. Une factorisation en transpositions d'une permutation est dite "primitive'' si ses facteurs sont ordonnés. Nous discutons du problème de l'énumération des factorisations primitives de permutations, et de sa place dans la hiérarchie des problèmes de factorisation précédemment étudiés. Nous présentons plusieurs formules énumérant certaines classes de factorisations primitives,et nous soulignons des connexions intéressantes avec les éléments Jucys-Murphy, les caractères des groupes symétriques, et les modèles de matrices.
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Barnes, George, Adrian Padellaro, and Sanjaye Ramgoolam. "Hidden symmetries and large N factorisation for permutation invariant matrix observables." Journal of High Energy Physics 2022, no. 8 (August 5, 2022). http://dx.doi.org/10.1007/jhep08(2022)090.
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Abstract Permutation invariant polynomial functions of matrices have previously been studied as the observables in matrix models invariant under SN, the symmetric group of all permutations of N objects. In this paper, the permutation invariant matrix observables (PIMOs) of degree k are shown to be in one-to-one correspondence with equivalence classes of elements in the diagrammatic partition algebra Pk (N). On a 4-dimensional subspace of the 13-parameter space of SN invariant Gaussian models, there is an enhanced O(N) symmetry. At a special point in this subspace, is the simplest O(N) invariant action. This is used to define an inner product on the PIMOs which is expressible as a trace of a product of elements in the partition algebra. The diagram algebra Pk (N) is used to prove the large N factorisation property for this inner product, which generalizes a familiar large N factorisation for inner products of matrix traces invariant under continuous symmetries.
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Nguenang, L. B., Emmanuel Kamgnia, and Bernard Philippe. "Localisation robuste et dénombrement de valeurs propres." Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées Volume 19 - 2015 - Special... (November 30, 2015). http://dx.doi.org/10.46298/arima.1983.
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International audience This article deals with the localization of eigenvalues of a large sparse and not necessarilysymmetric matrix in a domain of the complex plane. It combines two studies carried out earlier.The first work deals with the effect of applying small perturbations on a matrix, and referred to ase -spectrum or pseudospectrum. The second study describes a procedure for counting the numberof eigenvalues of a matrix in a region of the complex plain surrounded by a closed curve. The twomethods are combined in order to share the LU factorization of the resolvent, that intervenes in thetwo methods, so as to reduce the cost. The codes obtained are parallelized. L’article se consacre à la localisation de valeurs propres pour une grande matrice creuse, apriori non symétrique, dans un domaine du plan complexe. Il combine deux notions déjà étudiées. Lapremière précise l’effet de perturbations sur la matrice par la définition de e -spectre ou pseudospectre.La deuxième consiste à dénombrer les valeurs propres entourées par une courbe a priori donnéedans le plan complexe. A partir de travaux antérieurs, on combine ici les deux approches avec l’objectifde mettre en commun les factorisations LU de la résolvante nécessaire aux deux approches et d’endiminuer le nombre. Les codes obtenus sont parallélisés.
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Priya, K., and K. Rajkumar. "Hyperspectral image non-linear unmixing using joint extrinsic and intrinsic priors with L1/2-norms to non-negative matrix factorisation." Journal of Spectral Imaging, April 7, 2022. http://dx.doi.org/10.1255/jsi.2022.a4.
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Hyperspectral unmixing (HU) is one of the most active emerging areas in image processing that estimates the hyperspectral image’s endmember and abundance. HU enhances the quality of both spectral and spatial dimensions of the image by modifying the endmember and abundance parameters of the hyperspectral images. There are several HU algorithms available in the literature based on the linear mixing model (LMM) that deals with the microscopic contents of the pixels in the images. Non-negative matrix factorisation (NMF) is the prominent method widely used in LMMs that simultaneously estimates both the endmembers and abundances parameters along with some residual factors of the image to improve the quality of unmixing. In addition to this, the quality of the image is enhanced by incorporating some constraints to both endmember and abundance matrices with the NMF method. However, all the existing methods apply any of these constraints to the endmember and abundance matrices by considering the linearity features of the images. In this paper, we propose an unmixing model called joint extrinsic and intrinsic priors with L1/2 norms to non-negative matrix factorisation (JEIp L1/2-NMF) that applies multiple constraints simultaneously to both endmember and abundance matrices of the hyperspectral image to enhance its quality. Three main external and internal constraints such as minimum volume, sparsity and total variation are applied to both the endmembers and abundance parameters of the image. In addition, a L1/2-norms is imposed to extract good quality spectral data. Therefore, the proposed method enhances spatial as well as spectral data and considers the non-linearity of the pixels in the image by adding a residual term to the model. Performance of our proposed model is measured by using different quality measuring indexes on four benchmark public datasets and found that the proposed method shows outstanding performance compared to all the conventional baseline methods. Further, we also evaluated the performance of our method by varying the number of endmembers empirically and concluded that less than five endmembers provides high-quality spectral and spatial data during the unmixing process.
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Skandera, Mark. "The cluster basis $\mathbb{Z}[x_{1,1},…,x_{3,3}]." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AJ,..., Proceedings (January 1, 2008). http://dx.doi.org/10.46298/dmtcs.3598.
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International audience We show that the set of cluster monomials for the cluster algebra of type $D_4$ contains a basis of the $\mathbb{Z}$-module $\mathbb{Z}[x_{1,1},\ldots ,x_{3,3}]$. We also show that the transition matrices relating this cluster basis to the natural and the dual canonical bases are unitriangular and nonnegative. These results support a conjecture of Fomin and Zelevinsky on the equality of the cluster and dual canonical bases. In the event that this conjectured equality is true, our results also imply an explicit factorization of each dual canonical basis element as a product of cluster variables. Nous montrons que l'ensemble des monômes de l'algebre "cluster'' $D_4$ contient une base-$\mathbb{Z}$ pour le module $\mathbb{Z}[x_{1,1},\ldots ,x_{3,3}]$. Nous montrons aussi que les matrices transitoires qui relient cette base à la base canonique duale sont unitriangulaires. Ces résultats renforcent une conjecture de Fomin et de Zelevinsky sur l'égalité de ces deux bases. Si cette égalité s'avérait être vraie, notre résultat donnerait aussi une factorisation des éléments de la base canonique duale.
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Kamgnia, Emmanuel, and Louis Bernard Nguenang. "Some efficient methods for computing the determinant of large sparse matrices." Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées Volume 17 - 2014 - Special... (August 4, 2014). http://dx.doi.org/10.46298/arima.1968.
Full textAbstract:
International audience The computation of determinants intervenes in many scientific applications, as for example in the localization of eigenvalues of a given matrix A in a domain of the complex plane. When a procedure based on the application of the residual theorem is used, the integration process leads to the evaluation of the principal argument of the complex logarithm of the function g(z) = det((z + h)I - A)/ det(zI - A), and a large number of determinants is computed to insure that the same branch of the complex logarithm is followed during the integration. In this paper, we present some efficient methods for computing the determinant of a large sparse and block structured matrix. Tests conducted using randomly generated matrices show the efficiency and robustness of our methods. Le calcul de déterminants intervient dans certaines applications scientifiques, comme parexemple dans le comptage du nombre de valeurs propres d’une matrice situées dans un domaineborné du plan complexe. Lorsqu’on utilise une approche fondée sur l’application du théorème desrésidus, l’intégration nous ramène à l’évaluation de l’argument principal du logarithme complexe de lafonction g(z) = det((z + h)I − A)/ det(zI − A), en un grand nombre de points, pour ne pas sauterd’une branche à l’autre du logarithme complexe. Nous proposons dans cet article quelques méthodesefficaces pour le calcul du déterminant d’une matrice grande et creuse, et qui peut être transforméesous forme de blocs structurés. Les résultats numériques, issus de tests sur des matrices généréesde façon aléatoire, confirment l’efficacité et la robustesse des méthodes proposées.
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Alioli, Simone, Alessandro Broggio, and Matthew A. Lim. "Zero-jettiness resummation for top-quark pair production at the LHC." Journal of High Energy Physics 2022, no. 1 (January 2022). http://dx.doi.org/10.1007/jhep01(2022)066.
Full textAbstract:
Abstract We study the resummation of the 0-jettiness resolution variable $$ \mathcal{T} $$ T 0 for the top-quark pair production process in hadronic collisions. Starting from an effective theory framework we derive a factorisation formula for this observable which allows its resummation at any logarithmic order in the $$ \mathcal{T} $$ T 0 → 0 limit. We then calculate the $$ \mathcal{O} $$ O (αs) corrections to the soft function matrices and, by employing renormalisation group equation methods, we obtain the ingredients for the resummation formula up to next-to-next-to-leading logarithmic (NNLL) accuracy. We study the impact of these corrections to the 0-jettiness distribution by comparing predictions at different accuracy orders: NLL, NLL′, NNLL and approximate NNLL′ ($$ {\mathrm{NNLL}}_{\mathrm{a}}^{\prime } $$ NNLL a ′ ). We match these results to the corresponding fixed order calculations both at leading order and next-to-leading order for the t$$ \overline{t} $$ t ¯ +jet production process, obtaining the most accurate prediction of the 0-jettiness distribution for the top-quark pair production process at $$ {\mathrm{NNLL}}_{\mathrm{a}}^{\prime } $$ NNLL a ′ +NLO accuracy.