Jorma Merikoski - Academia.edu (original) (raw)
Papers by Jorma Merikoski
Education Research International, 2015
This study concerns the permanence of the basic arithmetical skills of Finnish students by invest... more This study concerns the permanence of the basic arithmetical skills of Finnish students by investigating how a group (N=463) of the eighth and eleventh year students and the university students of humanities perform in problems that are slightly modified versions of certain PISA 2003 mathematics test items. The investigation also aimed at finding out what the impact of motivation-related constructs, for example, students’ achievement goal orientations, is and what their perceived competence beliefs and task value on their performance in mathematics are. According to our findings, the younger students’ arithmetical skills have declined through the course of ten years but the older students’ skills have become generic to a greater extent. Further, three motivational clusters could be identified accounting for 7.5 per cent of students’ performance in the given assignments. These results are compatible with the outcomes of the recent assessments of the Finnish students’ mathematical ski...
The incorporation of nonnegativity constraints in image reconstruction problems is known to have ... more The incorporation of nonnegativity constraints in image reconstruction problems is known to have a stabilizing effect on solution methods. In this paper, we both demonstrate and provide an explanation of this phenomena when the image reconstruction problem of interest has least squares form. The benefits of using this natural constraint suggest the importance of incorporating a priori knowledge about solutions when possible. In fact, if this prior information is significantly strong, sophisticated likelihood functions and computational methods may not be necessary.
International Statistical Review
A k-circulant A k;n (1 = k = n-1) is an n-square matrix whose each row is obtained from the previ... more A k-circulant A k;n (1 = k = n-1) is an n-square matrix whose each row is obtained from the previous by k circular shifts to the right. Its first row (x 0 ,. .. , x n 1) is called the input. Nothing is said about the x i 's in the definitions, but I guess that they are real numbers. The matrix A k;n is a circulant C n if k = 1 and a reverse circulant RC n if k = n-1. (The term 'k-circulant' has also another meaning. It is often defined that a matrix is a k-circulant if it is obtained from C n by multiplying with k its all entries below the main diagonal.) If the sequence (x 1 ,. .. , x n 1) is palindromic, then C n is symmetric, denoted by SC n. Also, RC n is symmetric. The empirical spectral distribution function (ESD) of an n-square matrix is obtained by putting a mass 1=n at its each eigenvalue. The limiting spectral distribution function (LSD) of a sequence of n-square matrices is the weak limit (if it exists) of the sequence of their ESDs. In the case of random matrices, this limit is understood in some probabilistic sense. A description of some main points of Chapters 1-8 and 11 follows. The titles of Chapters 9 and 10 are informative enough. Chapter 1. The matrices A k;n , C n , RC n , and SC n are defined. Call them 'circulant-type matrices'. A formula for the eigenvalues of A k;n is given. This formula is fundamental in what follows (except Chapter 2). Chapter 2. The ESD and LSD are defined. A general technique to find the LSD of symmetric random matrices, based on the moment method, is introduced and applied to RC n and SC n when the input is i.i.d. (i.e. it consists of independent and identically distributed random variables).
Acta Universitatis Sapientiae, Mathematica
We define perpendicularity in an Abelian group G as a binary relation satisfying certain five axi... more We define perpendicularity in an Abelian group G as a binary relation satisfying certain five axioms. Such a relation is maximal if it is not a subrelation of any other perpendicularity in G. A motivation for the study is that the poset (𝒫, ⊆) of all perpendicularities in G is a lattice if G has a unique maximal perpendicularity, and only a meet-semilattice if not. We study the cardinality of the set of maximal perpendicularities and, on the other hand, conditions on the existence of a unique maximal perpendicularity in the following cases: G ≅ ℤ
International Journal of Mathematics and Mathematical Sciences, 2013
We give a set of axioms to establish a perpendicularity relation in an Abelian group and then stu... more We give a set of axioms to establish a perpendicularity relation in an Abelian group and then study the existence of perpendicularities in(ℤn,+)and(ℚ+,·)and in certain other groups. Our approach provides a justification for the use of the symbol⊥denoting relative primeness in number theory and extends the domain of this convention to some degree. Related to that, we also consider parallelism from an axiomatic perspective.
Mathematical Inequalities & Applications, 2001
Best possible bounds for real numbers λ 1 • • • λ n > 0 with prescribed sum a = λ 1 + • • • + λ n... more Best possible bounds for real numbers λ 1 • • • λ n > 0 with prescribed sum a = λ 1 + • • • + λ n and product d = λ 1 • • • λ n are presented. These bounds can be expressed algebraically only in certain special cases. In the general case, explicit bounds are found by using extra bounds. The results are applied to eigenvalue estimation, when the λ k 's are regarded as eigenvalues of an n by n matrix A , a = tr A , and d = det A. The case when the eigenvalues are real but not necessarily positive is also discussed. The bounds are compared with bounds using a and b = λ 2 1 + • • • + λ 2 n ; i.e, with eigenvalue bounds using tr A and tr A 2 .
Special Matrices
Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobeni... more Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobenius norm of the r-circulant matrix corresponding to g0, . . . , gn−1. We also give three lower bounds (with equality conditions) for the spectral norm of this matrix. For this purpose, we present three ways to estimate the spectral norm from below in general.
Czechoslovak Mathematical Journal
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Czechoslovak Mathematical Journal
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Kirjoitin viitteessä , että koska laajennuksessa kompleksilukujen joukosta C kvaternioiden joukko... more Kirjoitin viitteessä , että koska laajennuksessa kompleksilukujen joukosta C kvaternioiden joukkoon H menetetään kertolaskun vaihdantalaki, niin kvaternioilla ei ole kovin suurta merkitystä. Tällä en suinkaan vähätellyt kvaternioita (päinvastoin totesin niiden kiinnostavuuden), vaan lähinnä vertasin niiden merkitystä kompleksilukujen merkitykseen.
Lobachevskii Journal of Mathematics, 2010
Let A = (a ij ) be a nonnegative square matrix, let G = (g ij ) be its geometric symmetrization, ... more Let A = (a ij ) be a nonnegative square matrix, let G = (g ij ) be its geometric symmetrization, i.e., g ij = √ a ij a ji , and let ρ denote the Perron root. We present a simple proof for the well-known inequality ρ(A) ≥ ρ(G).
Linear Algebra and its Applications, 1992
Linear Algebra and Its Applications, May 1, 2003
The spread of an n × n matrix A with eigenvalues λ 1 , . . . , λ n is defined by spr A = max j,k ... more The spread of an n × n matrix A with eigenvalues λ 1 , . . . , λ n is defined by spr A = max j,k |λ j − λ k |. We prove that if A is normal, then
Journal of Inequalities in Pure and Applied Mathematics, 2004
Linear Algebra Appl, 1990
Linear Algebra and its Applications, 2003
The spread of an n × n matrix A with eigenvalues λ 1 , . . . , λ n is defined by spr A = max j,k ... more The spread of an n × n matrix A with eigenvalues λ 1 , . . . , λ n is defined by spr A = max j,k |λ j − λ k |. We prove that if A is normal, then
International Statistical Review, 2015
International Journal of Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics
Let A be a complex m ◊ n matrix. We find simple and good lower bounds for its spectral norm kAk =... more Let A be a complex m ◊ n matrix. We find simple and good lower bounds for its spectral norm kAk = max{kAxk | x 2 Cn, kxk = 1} by choosing x smartly. Here k · k applied to a vector denotes the Euclidean norm.
International Statistical Review, 2014
Readership: Graduate students, their teachers, and anybody doing research in statistics.
Education Research International, 2015
This study concerns the permanence of the basic arithmetical skills of Finnish students by invest... more This study concerns the permanence of the basic arithmetical skills of Finnish students by investigating how a group (N=463) of the eighth and eleventh year students and the university students of humanities perform in problems that are slightly modified versions of certain PISA 2003 mathematics test items. The investigation also aimed at finding out what the impact of motivation-related constructs, for example, students’ achievement goal orientations, is and what their perceived competence beliefs and task value on their performance in mathematics are. According to our findings, the younger students’ arithmetical skills have declined through the course of ten years but the older students’ skills have become generic to a greater extent. Further, three motivational clusters could be identified accounting for 7.5 per cent of students’ performance in the given assignments. These results are compatible with the outcomes of the recent assessments of the Finnish students’ mathematical ski...
The incorporation of nonnegativity constraints in image reconstruction problems is known to have ... more The incorporation of nonnegativity constraints in image reconstruction problems is known to have a stabilizing effect on solution methods. In this paper, we both demonstrate and provide an explanation of this phenomena when the image reconstruction problem of interest has least squares form. The benefits of using this natural constraint suggest the importance of incorporating a priori knowledge about solutions when possible. In fact, if this prior information is significantly strong, sophisticated likelihood functions and computational methods may not be necessary.
International Statistical Review
A k-circulant A k;n (1 = k = n-1) is an n-square matrix whose each row is obtained from the previ... more A k-circulant A k;n (1 = k = n-1) is an n-square matrix whose each row is obtained from the previous by k circular shifts to the right. Its first row (x 0 ,. .. , x n 1) is called the input. Nothing is said about the x i 's in the definitions, but I guess that they are real numbers. The matrix A k;n is a circulant C n if k = 1 and a reverse circulant RC n if k = n-1. (The term 'k-circulant' has also another meaning. It is often defined that a matrix is a k-circulant if it is obtained from C n by multiplying with k its all entries below the main diagonal.) If the sequence (x 1 ,. .. , x n 1) is palindromic, then C n is symmetric, denoted by SC n. Also, RC n is symmetric. The empirical spectral distribution function (ESD) of an n-square matrix is obtained by putting a mass 1=n at its each eigenvalue. The limiting spectral distribution function (LSD) of a sequence of n-square matrices is the weak limit (if it exists) of the sequence of their ESDs. In the case of random matrices, this limit is understood in some probabilistic sense. A description of some main points of Chapters 1-8 and 11 follows. The titles of Chapters 9 and 10 are informative enough. Chapter 1. The matrices A k;n , C n , RC n , and SC n are defined. Call them 'circulant-type matrices'. A formula for the eigenvalues of A k;n is given. This formula is fundamental in what follows (except Chapter 2). Chapter 2. The ESD and LSD are defined. A general technique to find the LSD of symmetric random matrices, based on the moment method, is introduced and applied to RC n and SC n when the input is i.i.d. (i.e. it consists of independent and identically distributed random variables).
Acta Universitatis Sapientiae, Mathematica
We define perpendicularity in an Abelian group G as a binary relation satisfying certain five axi... more We define perpendicularity in an Abelian group G as a binary relation satisfying certain five axioms. Such a relation is maximal if it is not a subrelation of any other perpendicularity in G. A motivation for the study is that the poset (𝒫, ⊆) of all perpendicularities in G is a lattice if G has a unique maximal perpendicularity, and only a meet-semilattice if not. We study the cardinality of the set of maximal perpendicularities and, on the other hand, conditions on the existence of a unique maximal perpendicularity in the following cases: G ≅ ℤ
International Journal of Mathematics and Mathematical Sciences, 2013
We give a set of axioms to establish a perpendicularity relation in an Abelian group and then stu... more We give a set of axioms to establish a perpendicularity relation in an Abelian group and then study the existence of perpendicularities in(ℤn,+)and(ℚ+,·)and in certain other groups. Our approach provides a justification for the use of the symbol⊥denoting relative primeness in number theory and extends the domain of this convention to some degree. Related to that, we also consider parallelism from an axiomatic perspective.
Mathematical Inequalities & Applications, 2001
Best possible bounds for real numbers λ 1 • • • λ n > 0 with prescribed sum a = λ 1 + • • • + λ n... more Best possible bounds for real numbers λ 1 • • • λ n > 0 with prescribed sum a = λ 1 + • • • + λ n and product d = λ 1 • • • λ n are presented. These bounds can be expressed algebraically only in certain special cases. In the general case, explicit bounds are found by using extra bounds. The results are applied to eigenvalue estimation, when the λ k 's are regarded as eigenvalues of an n by n matrix A , a = tr A , and d = det A. The case when the eigenvalues are real but not necessarily positive is also discussed. The bounds are compared with bounds using a and b = λ 2 1 + • • • + λ 2 n ; i.e, with eigenvalue bounds using tr A and tr A 2 .
Special Matrices
Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobeni... more Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobenius norm of the r-circulant matrix corresponding to g0, . . . , gn−1. We also give three lower bounds (with equality conditions) for the spectral norm of this matrix. For this purpose, we present three ways to estimate the spectral norm from below in general.
Czechoslovak Mathematical Journal
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Czechoslovak Mathematical Journal
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Kirjoitin viitteessä , että koska laajennuksessa kompleksilukujen joukosta C kvaternioiden joukko... more Kirjoitin viitteessä , että koska laajennuksessa kompleksilukujen joukosta C kvaternioiden joukkoon H menetetään kertolaskun vaihdantalaki, niin kvaternioilla ei ole kovin suurta merkitystä. Tällä en suinkaan vähätellyt kvaternioita (päinvastoin totesin niiden kiinnostavuuden), vaan lähinnä vertasin niiden merkitystä kompleksilukujen merkitykseen.
Lobachevskii Journal of Mathematics, 2010
Let A = (a ij ) be a nonnegative square matrix, let G = (g ij ) be its geometric symmetrization, ... more Let A = (a ij ) be a nonnegative square matrix, let G = (g ij ) be its geometric symmetrization, i.e., g ij = √ a ij a ji , and let ρ denote the Perron root. We present a simple proof for the well-known inequality ρ(A) ≥ ρ(G).
Linear Algebra and its Applications, 1992
Linear Algebra and Its Applications, May 1, 2003
The spread of an n × n matrix A with eigenvalues λ 1 , . . . , λ n is defined by spr A = max j,k ... more The spread of an n × n matrix A with eigenvalues λ 1 , . . . , λ n is defined by spr A = max j,k |λ j − λ k |. We prove that if A is normal, then
Journal of Inequalities in Pure and Applied Mathematics, 2004
Linear Algebra Appl, 1990
Linear Algebra and its Applications, 2003
The spread of an n × n matrix A with eigenvalues λ 1 , . . . , λ n is defined by spr A = max j,k ... more The spread of an n × n matrix A with eigenvalues λ 1 , . . . , λ n is defined by spr A = max j,k |λ j − λ k |. We prove that if A is normal, then
International Statistical Review, 2015
International Journal of Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics
Let A be a complex m ◊ n matrix. We find simple and good lower bounds for its spectral norm kAk =... more Let A be a complex m ◊ n matrix. We find simple and good lower bounds for its spectral norm kAk = max{kAxk | x 2 Cn, kxk = 1} by choosing x smartly. Here k · k applied to a vector denotes the Euclidean norm.
International Statistical Review, 2014
Readership: Graduate students, their teachers, and anybody doing research in statistics.