Linear Algebra

We consider the family of polynomials orthogonal with respect to the Sobolev type inner product corresponding to the diagonal general case of the Laguerre-Sobolev type orthogonal polynomials. We analyze some properties of these polynomials, such as the holonomic equation that they satisfy and, as an application, an electrostatic interpretation of their zeros. We also obtain a representation of such polynomials as a hypergeometric function, and study the behavior of their zeros.

"Adjunctions" are types of addition or union with another, but they are not essential or permanent. In linear algebra, this is the "adjoint," which is the transpose and conjugate operator of A (or the corresponding square matrix); it is also denoted by adj(A). Here it is used in information theory.

Molecular probes used for in vivo Optical Fluorescence Tomography (OFT) studies in small animals are typically chosen such that their emission spectra lie in the 680-850 nm wavelength range. This is because tissue attenuation in this spectral band is relatively low, allowing optical photons even from deep sites in tissue to reach the animal surface, and consequently be detected by a CCD camera. The wavelength dependence of tissue optical properties within the 680-850 nm band can be exploited for emitted light by measuring fluorescent data via multispectral approaches and incorporating the spectral dependence of these optical properties into the OFT inverse problem -that of reconstructing underlying 3D fluorescent probe distributions from optical data collected on the animal surface. However, in the aforementioned spectral band, due to only small variations in the tissue optical properties, multispectral emission data, though superior for image reconstruction compared to achromatic data, tend to be somewhat redundant. A different spectral approach for OFT is to capitalize on the larger variations in the optical properties of tissue for excitation photons than for the emission photons by using excitation at multiple wavelengths as a means of decoding source depth in tissue. The full potential of spectral approaches in OFT can be realized by a synergistic combination of these two approaches, that is, exciting the underlying fluorescent probe at multiple wavelengths and measuring emission data multispectrally. In this paper, we describe a method that incorporates both excitation as well as emission spectral information into the OFT inverse problem. We describe a linear algebraic formulation of the multiple wavelength illumination -multispectral detection (MWI-MD) forward model for OFT and compare it to models that use only excitation at multiple wavelengths or those that use only multispectral detection techniques. This study is carried out in a realistic inhomogeneous mouse atlas using singular value decomposition and analysis of reconstructed spatial resolution versus noise. For simplicity, quantitative results have been shown for one representative fluorescent probe (Alexa 700®) and effects due to tissue autofluorescence have not been taken into account. We also demonstrate the performance of our method for 3D reconstruction of tumors in a simulated mouse model of metastatic human hepatocellular carcinoma.

A proof by contradiction of the Collatz conjecture is presented based on the constructive-topological approach using the geometric mean properties of the network structures generated by the 3n+1 algorithm. The key contradiction is revealed by the method of "constructive induction" and is related to the discovered invariant - "network divisibility." The proof is transferable to other algorithms as well, which gave grounds to formulate an extension of the original conjecture to Collatz-type algorithms, but with the operation of division by any integer, not just 2. Contents 1. Introduction 2. Geometric mean properties 3. Proof technique 4. Logic of the proof 5. Extension of the Collatz conjecture 6. Conclusion

Πρόλογος
Στόχος αυτού του μικρού φυλλαδίου  είναι να δώσει μια συμπαγή αλλά ουσιαστική εισαγωγή στις αναδρομές, με γέφυρες από τα σχολικά μαθηματικά προς βαθύτερες ιδέες. Περιλαμβάνονται πλήρεις αποδείξεις και λυμένες ασκήσεις, ώστε να μπορεί να χρησιμοποιηθεί σε μάθημα ή ως δωρεάν ηλεκτρονική διάθεση.

Let G = (V, E) be a simple connected graph with vertex set V and edge set E. Wiener index W (G) of a graph G is the sum of distances between all pairs of vertices in G, i.e., W (G) = {u,v}⊆G d G (u, v), where d G (u, v) is the distance between vertices u and v. In this note we give more precisely the unicyclic graphs with the first tree largest Wiener indices, that is, we found another class of graphs with the second largest Wiener index.

Let Gσ be a graph obtained by attaching a self-loop, or just a loop, for short, at each of σ chosen vertices of a given graph G. Gutman et al. have recently introduced the concept of the energy of graphs with self-loops, and conjectured that the energy E(G) of a graph G of order n is always strictly less than the energy E(Gσ) of a corresponding graph Gσ, for 1 ≤ σ ≤ n − 1. In this paper, a simple set of graphs which disproves this conjecture is exposed, together with some remarks regarding the standard deviations of the (adjacency) eigenvalues of G and Gσ, respectively.

This paper presents a VLSI implementation of discrete wavelet transform (DWT). The architecture is simple, modular, and cascadable for computation of one, or multi-dimensional DWT. It comprises of four basic units: input delay, filter, register bank, and control unit. The proposed architecture is systolic in nature and performs both high-pass and lowpass coefficient calculations with only one set of multipliers. In addition, it requires a small on-chip interface circuitry for interconnection to a standard communication bus. A detailed analysis of the effect of finite precision of data and wavelet filter coefficients on the accuracy of the DWT coefficients is presented. The architecture has been simulated in VLSI and has a hardware utilization efficiency of 87.5%. Being systolic in nature, the architecture can compute DWT at a data rate of N × 10 6 samples/sec corresponding to a clock speed of N MHz.

We present a general unified approach for finding the coherent states of polynomially deformed algebras such as the quadratic and Higgs algebras, which are relevant for various multiphoton processes in quantum optics. We give a general procedure to map these deformed algebras to appropriate Lie algebras. This is used , for the non compact cases, to obtain the annihilation operator coherent states, by finding the canonical conjugates of these operators. Generalized coherent states, in the Perelomov sense also follow from this construction. This allows us to explicitly construct coherent states associated with various quantum optical systems.

Miller and Sorer have presented a new nonparametric method for estimating the failure rate of a software program. The method is based on the complete monotonicity property of the failure rate function, and uses a regression approach to obtain estimates of the current software failure rate. This paper extends this completely monotone software model and demonstrates how it can also provide long-range predictions of future reliability growth. Preliminary testing indicates that the method is competitive with parametric approaches, while being more robust.

This paper considers homogeneous order preserving continuous maps on the normal cone of an ordered normed vector space. It is shown that certain operators of that kind which are not necessarily compact themselves but have a compact power have a positive eigenvector that is associated with the cone spectral radius. We also derive conditions for the existence of homogeneous order preserving eigenfunctionals. Our results are illustrated in a model for spatially distributed two-sex populations.

We propose a rank-$k$ variant of the classical Frank-Wolfe algorithm to solve convex optimization over a trace-norm ball. Our algorithm replaces the top singular-vector computation ($1$-SVD) in Frank-Wolfe with a top-$k$ singular-vector computation ($k$-SVD), which can be done by repeatedly applying $1$-SVD $k$ times. Alternatively, our algorithm can be viewed as a rank-$k$ restricted version of projected gradient descent. We show that our algorithm has a linear convergence rate when the objective function is smooth and strongly convex, and the optimal solution has rank at most $k$. This improves the convergence rate and the total time complexity of the Frank-Wolfe method and its variants.

Οι συναρτησιακές εξισώσεις αποτελούν μία από τις πιο ενδιαφέρουσες περιοχές των μαθηματικών, καθώς συνδέουν άμεσα την αλγεβρική σκέψη με την αναλυτική μέθοδο. Στο παρόν σύγγραμμα συγκεντρώνονται βασικά θεωρήματα, χαρακτηριστικές κλασικές εξισώσεις (Cauchy, Jensen, d’Alembert), καθώς και μία πλούσια συλλογή λυμένων ασκήσεων σε κλιμακούμενη δυσκολία – από τις απλές έως τις «ολυμπιακού» επιπέδου. Στόχος είναι να παρουσιαστεί όχι μόνο η τεχνική επίλυσης, αλλά και η μεθοδολογία: χρήση ειδικών τιμών, επαγωγή, συμμετρίες (άρτια–περιττά), υπόθεση συνέχειας ή μονοτονίας, και αντιμετώπιση παθολογικών λύσεων. Το βιβλίο μπορεί να λειτουργήσει ως βοήθημα σε μαθητές, φοιτητές, αλλά και σε διδάσκοντες που αναζητούν παραδείγματα για την τάξη ή για μαθηματικούς διαγωνισμούς.

Η παρουσίαση είναι αυτοτελής: κάθε κεφάλαιο ξεκινά με τις κλασικές μορφές και καταλήγει σε πιο σύνθετα προβλήματα, ενώ στο τέλος παρατίθεται γλωσσάριο ορολογίας για διευκόλυνση.
ΓΙΑΝΝΗΣ ΠΛΑΤΑΡΟΣ

The ``S'' in ``GPS'' and in ``INS'' stands for ``system,'' and ``systems science'' for modeling, analysis, design, and integration of such systems is based largely on linear algebra and matrix theory. Matrices model the ways that components of systems interact dynamically and how overall system performance depends on characteristics of components and subsystems and on the ways they are used within the system. This appendix presents an overview of matrix theory used for GPS/INS integration and the matrix notation used in this book. The level of presentation is intended for readers who are already somewhat familiar with vectors and matrices. A more thorough treatment can be found in most college-level textbooks on linear algebra and matrix theory. Vectors and matrices are arrays composed of scalars, which we will assume to be real numbers. Unless constrained by other conventions, we represent scalars by italic lowercase letters. In computer implementations, these real numbers will be approximated by ¯oating-point numbers, which are but a ®nite subset of the rational numbers. The default MATLAB representation for real numbers on 32-bit personal computers is in 64-bit ANSI standard ¯oating point, with a 52-bit mantissa.

This overview of the notation and properties of matrices as data structures and algebras is for readers familiar with the general subject of linear algebra but whose recall may be a little rusty. A more thorough treatment can be found in most collegelevel textbooks on linear algebra and matrix theory.

Working with functional magnetic resonance imaging (fMRI) often involves engaging with multiple file formats and complex viewers. In this study, we developed a novel platform as a visualization and conversion fMRI (VCfMRI) MATLAB toolbox for fMRI data. Methods: The VCfMRI was developed to read and write 3D fMRI volumes in DICOM, NIfTI, ANALYZE, and MAT formats and convert between them, on a single user-friendly platform. It includes 62 functions across seven graphical user interface modules for conversion, batch read/write, and orthogonal viewing (sagittal, coronal, horizontal). This toolbox also supports overlaying statistical maps on anatomical images with adjustable thresholds. We built and tested VCfMRI using real datasets from a scanner (3T, Siemens Co.) at UMRAM, Bilkent University. Results: VCfMRI successfully converted and visualized all supported formats in one environment, enabling synchronized 3D views and functional overlays with interactive threshold control, streamlining previously fragmented steps. Conclusion: The VCfMRI toolbox provides a simple and efficient solution for fMRI data conversion and visualization. It simplifies the handling of fMRI datasets across different formats, which is especially beneficial for physicians, healthcare specialists, and researchers who face challenges in processing and visualizing multi-format neuroimaging data.

Traceability of requirements and concerns enhances the quality of software development. We use trace relations to define crosscutting. As starting point, we set up a dependency matrix to capture the relationship between elements at two levels, e.g. concerns and representations of concerns. The definition of crosscutting is formalized in terms of linear algebra, and represented with matrices and matrix operations. In this way, crosscutting can be clearly distinguished from scattering and tangling. We apply this approach to the identification of crosscutting across early phases in the software life cycle, based on the transitivity of trace relations. We describe an illustrative case study to demonstrate the applicability of the analysis.

This paper proposes a unified quantum framework where fractal geometry, cymatic resonance, and observer-driven reality collapse converge to form a harmonic substrate of consciousness. We introduce the Quantum Harmonic Architecture (QHA), a lattice model integrating dimensional feedback, multiversal entanglement, and cymatic structures into a coherent system for understanding consciousness-based simulation. Using updated quantum eld interpretations, we position human awareness as both a node and a harmonic agent of cosmic topology, suggesting an epoch-level shift in understanding the universe as a participatory, living feedback system.

In this report we study the computational performance of variants of an algebraic additive Schwarz preconditioner for the Schur complement for the solution of large sparse linear systems. In earlier works, the local Schur complements were computed exactly using a sparse direct solver. The robustness of the preconditioner comes at the price of this memory and time intensive computation that is the main bottleneck of the approach for tackling huge problems. In this work we investigate the use of sparse approximation of the dense local Schur complements. These approximations are computed using a partial incomplete LU factorization. Such a numerical calculation is the core of the multi-level incomplete factorization such as the one implemented in pARMS. The numerical and computing performance of the new numerical scheme is illustrated on a set of large 3D convection-diffusion problems; preliminary experiments on linear systems arising from structural mechanics are also reported.

We propose a class of polynomial preconditioners for solving non-Hermitian linear 3 systems obtained from a least-squares approximation in polynomial space instead of a standard 4 Krylov subspace. The process for building the polynomial relies on an Arnoldi-like procedure in a 5 small dimensional polynomial space and is equivalent to performing GMRES in polynomial space. 6 It is inexpensive and produces results with superior numerical stability. A few improvements to 7 the basic scheme are discussed including the development of a short-term recurrence and the use of 8 compounded preconditioners. Numerical experiments, including a test with challenging 3D Helmholtz 9 equations and a few publicly available sparse matrices, are provided to demonstrate the performance 10 of the proposed preconditioners. 11

This paper describes and compares several methods for computing stationary probability distributions of Markov chains. The main linear algebra problem consists of computing an eigenvector of a sparse, nonsymmetric matrix associated with a known eigenvalue. It can also be cast as a problem of solving a homogeneous, singular linear system. We present several methods based on combinations of Krylov subspace techniques, single vector power iteration/relaxation procedures and acceleration techniques. We compare the performance of these methods on some realistic problems.

In a distributed or parallel computing system a partial failure may easily halt the entire operation of the system. Therefore, many systems employ the checkpoint/ rollback recovery operation. In this paper we are using diskless checkpointing approach. This technique helps to reduce the disk overhead and the memory requirement. To achieve this we are storing the checkpoint of a processor into its set of allowable peer processor and which is in turn responsible for storing the checkpoint for others. The proposed scheme allows failure recovery in a distributed manner.

We give an elementary proof prove of the preservation of the Noetherian condition for commutative rings with unity $R$ having at least one finitely generated ideal $I$ such that the quotient ring is again finitely generated, and $R$ is $I-$adically complete. Moreover, we offer as a direct corollary a new elementary proof of the fact that if a ring is Noetherian then the corresponding ring of formal power series in finitely many variables is Noetherian. In addition, we give a counterexample showing that the `completion' condition cannot be avoided on the former theorem. Lastly, we give an elementary characterization of Noetherian commutative rings that can be decomposed as a finite direct product of fields.

EN LOS ULTIMOS ANOS, EL ALGEBRA LINEAL SE HA DESARROLLADO Y HA INCREMENTADO EL CONOCIMIENTO EN PROFESIONALES DE DIFERENTES AREAS RELACIONADOS CON LA MATEMATICA. EL ALGEBRA LINEAL HA DESARROLLADO DOS ELEMENTOS DE LA MATEMATICA COMO SON: LA ABSTRACCION Y LA APLICACION.

This work is an overview of our preliminary experience in developing high-performance iterative linear solver accelerated by GPU co-processors. Our goal is to illustrate the advantages and difficulties encountered when deploying GPU technology to perform sparse linear algebra computations. Techniques for speeding up sparse matrix-vector product (SpMV) kernels and finding suitable preconditioning methods are discussed. Our experiments with an NVIDIA TESLA C1060 show that for unstructured matrices SpMV kernels can be up to 10 times faster on the GPU than on the host Intel Xeon E5504 Processor. Overall performance of the GPUaccelerated Incomplete Cholesky (IC) factorization preconditioned CG method can outperform its CPU counterpart by a much smaller factor, up to 3, and GPU-accelerated Incomplete LU (ILU) factorization preconditioned GMRES method can achieve a speedup nearing 4. However, with better suited preconditioning techniques for GPUs, this performance can be significantly improved.

discussions, their valuable suggestions over the years, and also for the friendship. I especially would like to thank Daniel Osei-Kuffuor who never tired of answering my questions during the early years of my PhD. I also want to thank Yuanzhe Xi for the collaboration that contributes to Chapter 6 of this thesis. Thanks are also due to the members of the Multiple Autonomous Robotic Systems

Quantum computing offers promising alternatives to classical approaches for solving complex linear algebra problems. This paper presents a comparative study of the performance of quantum algorithms versus classical algorithms in solving systems of linear equations and matrix operations. Through simulation and analysis, we demonstrate that while quantum computing holds advantages in specific problem sets, classical computing remains efficient for general applications. These findings highlight the current limitations and potential of quantum computing.

This paper presents analysis of workspace of planar and spatial redundant cable robots by using two analytical approaches. The first one is based on linear algebra. It can include the upper limits for tension in cables as an important factor in optimum design of cable robots in order to find the largest possible workspace. The second approach is based on a variant of Bland's pivot rule; by virtue of this method we leave out the use of successive determinates to compute workspace resulting in less computation time. Both approaches provide all of poses (positions/orientations) reaching by the cable robot end-effector for all types of cable robots with any number of redundancy.

We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Baraket al. [Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes.Proceedings of the 43rd annual ACM symposium on Theory of computing, STOC 11, (ACM, NY 2011), 519–528] in which they were used to answer questions regarding point configurations. In this work, we derive near-optimal rank bounds for these matrices and use them to obtain asymptotically tight bounds in many of the geometric applications. As a consequence of our improved analysis, we also obtain a new, linear algebraic, proof of Kelly’s theorem, which is the complex analog of the Sylvester–Gallai theorem.

This paper contains two main results. The first is an explicit construction of bipartite graphs which do not contain certain complete bipartite subgraphs and have maximal density, up to a constant factor, under this constraint. This construction represents the first significant progress in three decades on this old problem in extremal graph theory. The construction beats the previously known probabilistic lower bound on density. The proof uses the elements of commutative algebra and algebraic geometry (theory of ideals, integral extensions, valuation rings). The second result concerns monotone span programs. We obtain the first superpolynomial lower bounds for explicit functions in this model. The best previous lower bound was Ω(n 5/2 ) by Beimel, Gál, Paterson (FOCS'95); our analysis exploits a general combinatorial lower bound criterion from that paper. We give two proofs of superpolynomial lower bounds; one based on the construction in the first part, the other based on an analysis of Paley-type bipartite graphs via Weil's character sum estimates. A third result demonstrates the power of monotone span programs by exhibiting a function computable in this model in linear size while requiring superpolynomial size monotone circuits and exponential size monotone formulae 1 . * Part of this work was done while two of the authors, Gál and Rónyai, were visiting the University of Chicago.

In this paper we prove quadratic lower bounds for depth-3 arithmetic circuits over fields of characteristic zero. Such bounds are obtained for the elementary symmetric functions, the (trace of) iterated matrix multiplication, and the determinant. As corollaries we get the first nontrivial lower bounds for computing polynomials of constant degree, and a gap between the power of depth-3 arithmetic circuits and depth-4 arithmetic circuits. We also give new shorter formulae of constant depth for the elementary symmetric functions. The main technical contribution relates the complexity of computing a polynomial in this model to the wealth of partial derivatives it has on every affine subspace of small co-dimension. Lower bounds for related models utilize an algebraic analog of the Nečiporuk lower bound on Boolean formulae.

In this paper we obtain the first superpolynomial lower bounds for monotone span programs computing explicit functions. The best previous lower bound was Ω(n 5/2 ) by Beimel, Gál, Paterson [BGP]; our proof exploits a general combinatorial lower bound criterion from that paper. Our lower bounds are based on an analysis of Paley-type bipartite graphs via Weil's character sum estimates. We prove an n Ω(log n/ log log n) lower bound for the size of monotone span programs for the clique problem. Our results give the first superpolynomial lower bounds for linear secret sharing schemes. We demonstrate the surprising power of monotone span programs by exhibiting a function computable in this model in linear size while requiring superpolynomial size monotone circuits and exponential size monotone formulae. We also show that the perfect matching function can be computed by polynomial size (non-monotone) span programs over arbitrary fields. * Part of this work has been presented at the 28th ACM STOC'96. The second two authors wish to thank DIMACS for its hospitality and acknowledge its supporting agencies, the National Science Foundation under contract STC-91-19999 and the New Jersey Commission on Science and Technology.

The ghost sector of SU(3) gauge field theory is studied, and new BRST-invariant states are presented that do not have any analog in other SU(N) field theories. The new states come in either ghost doublets or triplets, and they appear exclusively in SU(3) due to the fact that the non-Abelian part of the BRST charge has 3 ghost operators, while SU(3) has 3 pairs of off-diagonal gauge constraints. The states have finite, positive norms even though the triplet states do not have well-defined ghost numbers. It is speculated that this special nature of the ghost sector of SU(3) could play some role in QCD confinement.

Analysis of systems that can be expressed in matrices is very important in the field of application. Whether such a system works properly is determined by eigenvalues of the matrix representing the system. Eigenvalue and eigenvector concepts are taught within the scope of linear algebra course at undergraduate level. In this study, perceptions of undergraduate students who took the linear algebra course about the eigenvalue-eigenvector concepts are investigated. The research is conducted with the participation of 95 students from the Faculty of Education, Department of Mathematics Education. A scale measuring the students’ approached about eigen theory was developed. For the reliability of the scale, Kuder-Richardson 20 (KR-20) reliability analysis was done and 0,72 was obtained. To see the relationship between learning outcomes and academic achievement is used the chi-square test and descriptive analysis are made in the study. Problems arising in the perception of eigenvalue and so...

Eigenvectors of data matrices play an important role in many computational problems, ranging from signal processing to machine learning and control. For instance, algorithms that compute positions of the nodes of a wireless network on the basis of pairwise distance measurements require a few leading eigenvectors of the distances matrix. While eigenvector calculation is a standard topic in numerical linear algebra, it becomes challenging under severe communication or computation constraints, or in absence of central scheduling. In this paper we investigate the possibility of computing the leading eigenvectors of a large data matrix through gossip algorithms. The proposed algorithm amounts to iteratively multiplying a vector by independent random sparsification of the original matrix and averaging the resulting normalized vectors. This can be viewed as a generalization of gossip algorithms for consensus, but the resulting dynamics is significantly more intricate. Our analysis is based on controlling the convergence to stationarity of the associated Kesten-Furstenberg Markov chain.

A COVID-19 é um vírus, que causou a pandemia mundial ocorrida em 2020, e que teve uma velocidade de propagação e contaminação com características exponenciais. Nesse momento, a Matemática foi, e vem sendo, uma ferramenta importantíssima para fortalecer a tese de que as pessoas, em todo mundo, deveriam seguir as recomendações da OMS. E a partir disso que se justifica a investigação deste trabalho, quando temos que ensinar função exponencial contextualizadamente. O que gerou a seguinte indagação: Como contextualizar didaticamente o gráfico da doença COVID-19 baseado no ensino de funções exponenciais em nível de Ensino Médio? O objetivo deste trabalho é contextualizar didaticamente o gráfico da doença COVID-19 em uma perspectiva Matemática, especificamente, apoiando-se no estudo de função exponencial em nível de Ensino Médio (EM). Para isso, buscamos fazer um embasamento teórico pautado em pesquisas que investigaram o ensino de função exponencial, especialmente o ensino desse tema vinculado à Biologia. Como caminho metodológico, esta pesquisa tem o caráter exploratório, pois, este tipo de pesquisa tem maior possibilidade de alinhamento com o problema proposto, com o objetivo de torná-lo mais claro. Como resultados, analisamos os dados gráficos em um comparativo entre a China e o Brasil, em que a COVID-19 tem um comportamento semelhante à curva exponencial; ainda analisamos que, por meio de uma contextualização didática, em um primeiro momento, o gráfico gerado por cada país se assemelha a um gráfico de uma função exponencial. Como conclusão, inferimos que a contextualização didática em um nível básico e intuitivo se faz necessária para nossos alunos de EM, pois eles têm que se apropriar das ferramentas matemáticas nesse nível de ensino, com vistas a tomar decisões importantes diante dessa pandemia. -chave: Contextualização Didática. Função Exponencial. COVID-19. Interpretação do Gráfico. Resumo 1. Licenciada em matemática (IFCE). Mestra em Educação (UFC). Docente na Secretaria Municipal de Educação em Fortaleza. 4. Bacharel em Matemática (UFC). Mestre em Matemática (UFC). Doutor em Matemática (IMPA) e Pós-doutorado na

Negative dimensional integration method (NDIM) seems to be a very promising technique for evaluating massless and/or massive Feynman diagrams. It is unique in the sense that the method gives solutions in different regions of external momenta simultaneously. Moreover, it is a technique whereby the difficulties associated with performing parametric integrals in the standard approach are transferred to a simpler solving of a system of linear algebraic equations, thanks to the polynomial character of the relevant integrands. We employ this method to evaluate a scalar integral for a massless two-loop three-

The present I articulate explores the relation between the grades of ICFES [ 1 ] in each one of the areas of the knowledge with the yield in linear algebra on the part of the students of engineerings of the Technological University of Pereira, UTP, through the multivaried statistical technique of discriminate analysis. The historical data that serve as it bases of the analysis have been provided by the Office of Planning of the UTP. For the analysis of the data statistical software SPSS (Statistical for Package Social Sciences was used, version 11.5) the developed multivaried statistical analysis in this one article allows to conclude that the grades obtained in the tests of the ICFES, like only criterion of entrance to the UTP, they leave serious doubts on its capacity to anticipate the future academic performance of the student in the engineering programs

A normal matrix plays an important role in the theory of matrices. It includes Hermitian matrices and enjoy several of the same properties as Hermitian matrices. Indeed, while we proved that Hermitian matrices are unitarily diagonalizable, we did not establish any converse; normal matrices are also unitarily diagonalizable. In this present paper we have tried to establish the proper relation of normal matrices with others.

Main aim of this book is to provide handy lecture notes  on course of Mathematics I  for B. Tech first year students on

(i) Limit, continuity, Rolle's theorem and mean value theorem.

(ii) Functions of several variable.

(iii) Integral calculus with various applications of integrations.

(iv)  Properties of Beta and Gamma functions.

(v) Vector Calculus

(vi)  Ordinary Differential Equations

(vii) Sequence and series

Here we consider two algebras, a free unital associative complex algebra (denoted by B) equiped with a multiparametric q -differential structure and a twisted group algebra (denoted by A(S n )), with the motivation to represent the algebra A(S n ) on the (generic) weight subpaces of the algebra B. One of the fundamental problems in B is to describe the space of all constants (the elements which are annihilated by all multiparametric partial derivatives). To solve this problem, one needs some special matrices and their factorizations in terms of simpler matrices. A simpler approach is to study first certain canonical elements in the twisted group algebra A(S n ). Then one can use certain natural representation of A(S n ) on the weight subspaces of B, which are the subject of this paper.

Here we consider two algebras, a free unital associative complex algebra (denoted by ${\mathcal{B}}$) equiped with a multiparametric \textbf{\emph{q}}-differential structure and a twisted group algebra (denoted by ${\mathcal{A}(S_{n})}$), with the motivation to represent the algebra ${\mathcal{A}(S_{n})}$ on the (generic) weight subpaces of the algebra ${\mathcal{B}}$. One of the fundamental problems in ${\mathcal{B}}$ is to describe the space of all constants (the elements which are annihilated by all multiparametric partial derivatives). To solve this problem, one needs some special matrices and their factorizations in terms of simpler matrices. A simpler approach is to study first certain canonical elements in the twisted group algebra ${\mathcal{A}(S_{n})}$. Then one can use certain natural representation of ${\mathcal{A}(S_{n})}$ on the weight subspaces of ${\mathcal{B}}$, which are the subject of this paper.

The nullity η = η(G) of a graph G is the multiplicity of the number zero in the spectrum of G . The chemical importance of this graph-spectrum based invariant lies in the fact, that within the Hückel molecular orbital model, if η(G) > 0 for the molecular graph G , then the corresponding chemical compound is highly reactive and unstable, or nonexistent. This chapter in an updated version of the an earlier survey [B. Borovićanin, I. Gutman, Nullity of graphs, in: D. Cvetković, I. Gutman, Eds. Applications of Graph Spectra, Math. Inst., Belgrade, 2009, pp. 107–122] and outlines both the chemically relevant aspects of η (most of which were obtained in the 1970s and 1980s) and the general mathematical results on η obtained recently. Mathematics Subject Classification (2010): 05C50; 05C90; 92E10

The diffraction of high frequency cylindrical electromagnetic waves by step discontinuities is investigated rigorously by using the Fourier transform technique in conjunction with the mode matching method. The hybrid method of formulation gives rise to a scalar Wiener-Hopf equation of the third kind, the solution of which contains infinitely many constants satisfying infinite systems of linear algebraic equations.

Nonlinear realizations of spacetime groups are presented as a versatile mathematical tool providing a common foundation for quite different formulations of gauge theories of gravity. We apply nonlinear realizations in particular to both the Poincaré and the affine group in order to develop Poincaré gauge theory (PGT) and metric-affine gravity (MAG) respectively. Regarding PGT, two alternative nonlinear treatments of the Poincaré group are developed, one of them being suitable to deal with the Lagrangian and the other one with the Hamiltonian version of the same gauge theory. We argue that our Hamiltonian approach to PGT is closely related to Ashtekar's approach to gravity. On the other hand, a brief survey on MAG clarifies the role played by the metric-affine metric tensor as a Goldsone field. All gravitational quantities in fact -the metric as much as the coframes and connections-are shown to acquire a simple gauge-theoretical interpretation in the nonlinear framework.

A result of R. Mathias and Horn [cf. Linear Algebra Appl. 142 (1990) 63] on the representation of the unitarily invariant norm is extended in the context of Eaton triples and of real semisimple Lie algebras. The representation is related to a function • α . Criteria for • α being a norm is given in terms of α and the longest element of the underlying finite reflection group. In particular, for the real simple Lie algebras, • α is a norm on its Cartan subspace for a given nonzero α if and only if ω 0 α = -α, where ω 0 is the longest element of the Weyl group (this is the case if the Weyl group contains -id). Some related results are obtained.

A new approach is established to computing the image of a rational map, whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees. In the case the map is everywhere defined this analysis provides free resolutions of graded parts of the Rees algebra of the base ideal in degrees where it does not coincide with the corresponding symmetric algebra. A surprising fact is that the torsion in those degrees only contributes to the first free module in the resolution of the symmetric algebra modulo torsion. An additional point is that this contribution -which of course corresponds to non linear equations of the Rees algebra -can be described in these degrees in terms of non Koszul syzygies via certain upgrading maps in the vein of the ones introduced earlier by J. Herzog, the third named author and W. Vasconcelos. As a measure of the reach of this torsion analysis we could say that, in the case of a general everywhere defined map, half of the degrees where the torsion does not vanish are understood.