Explicit solutions of the matrix equationAX−XB=C

1974 ◽  
Vol 23 (2-3) ◽  
pp. 245-257 ◽  
Author(s):  
John Jones
Keyword(s):  
Explicit Solutions ◽  
The Matrix

scholarly journals Algebraically Solvable Problems: Describing Polynomials as Equivalent to Explicit Solutions

10.37236/734 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Combinatorial Nullstellensatz ◽  
Algebraic Solution ◽  
Explicit Solutions ◽  
Polynomial Map ◽  
Combinatorial Number Theory ◽  
Special Cases ◽  
The Matrix ◽  
Grid Points ◽  
Algebraic Solutions ◽  
Solvable Problems

The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi's Combinatorial Nullstellensatz. On its own, it is a result about polynomials, providing some information about the polynomial map $P|_{\mathfrak{X}_1\times\cdots\times\mathfrak{X}_n}$ when only incomplete information about the polynomial $P(X_1,\dots,X_n)$ is given.In a very general working frame, the grid points $x\in \mathfrak{X}_1\times\cdots\times\mathfrak{X}_n$ which do not vanish under an algebraic solution – a certain describing polynomial $P(X_1,\dots,X_n)$ – correspond to the explicit solutions of a problem. As a consequence of the coefficient formula, we prove that the existence of an algebraic solution is equivalent to the existence of a nontrivial solution to a problem. By a problem, we mean everything that "owns" both, a set ${\cal S}$, which may be called the set of solutions; and a subset ${\cal S}_{\rm triv}\subseteq{\cal S}$, the set of trivial solutions.We give several examples of how to find algebraic solutions, and how to apply our coefficient formula. These examples are mainly from graph theory and combinatorial number theory, but we also prove several versions of Chevalley and Warning's Theorem, including a generalization of Olson's Theorem, as examples and useful corollaries.We obtain a permanent formula by applying our coefficient formula to the matrix polynomial, which is a generalization of the graph polynomial. This formula is an integrative generalization and sharpening of:1. Ryser's permanent formula.2. Alon's Permanent Lemma.3. Alon and Tarsi's Theorem about orientations and colorings of graphs.Furthermore, in combination with the Vigneron-Ellingham-Goddyn property of planar $n$-regular graphs, the formula contains as very special cases:4. Scheim's formula for the number of edge $n$-colorings of such graphs.5. Ellingham and Goddyn's partial answer to the list coloring conjecture.

scholarly journals CRITICAL SCALING IN THE MATRIX MODEL ON A BETHE TREE

1994 ◽  
Vol 09 (21) ◽  
pp. 1963-1973 ◽  
Author(s):  
D.V. BOULATOV
Keyword(s):  
Matrix Model ◽  
Explicit Solutions ◽  
Critical Scaling ◽  
Edge Singularity ◽  
Large N ◽  
The Matrix

The matrix model with a Bethe tree embedding space (coincides at large N with the Kazakov-Migdal “induced QCD” model1) is investigated. We further elaborate the Riemann-Hilbert approach of Ref. 2 assuming certain holomorphic properties of the solution. The critical scaling (an edge singularity of the density) is found to be [Formula: see text][Formula: see text] arccos D, for |D|<1, and [Formula: see text] arccos [Formula: see text] for D>1. Explicit solutions are constructed at D=1/2 and D=∞.

The Interaction between a Dislocation and Circular Inhomogeneity in 1D Hexagonal Quasicrystals

2010 ◽  
Vol 34-35 ◽  
pp. 429-434
Author(s):  
Ya Qun Hu ◽  
Ping Xia ◽  
Ke Xiang Wei
Keyword(s):  
Elastic Constant ◽  
Complex Potential ◽  
Image Force ◽  
Potential Method ◽  
Complex Potentials ◽  
Explicit Solutions ◽  
Circular Inhomogeneity ◽  
The Matrix ◽  
Complex Potential Method ◽  
1D Hexagonal Quasicrystals

The interaction between a dislocation and circular inhomogeneity in 1D hexagonal quasicrystals is investigated. By using the complex potential method, explicit solutions of complex potentials are obtained. The image force acting on the dislocation are also derived. The results show that the interface attracts the dislocation inside both the matrix and the inhomogeneity under most condition. The attraction increase with the increase of the elastic constant of phason field and the phonon-phason coupling elastic constant.

scholarly journals Re-nnd SOLUTIONS OF THE MATRIX EQUATION AXB=C

2008 ◽  
Vol 84 (1) ◽  
pp. 63-72 ◽  
Author(s):  
DRAGANA S. CVETKOVIĆ-ILIĆ
Keyword(s):  
Inverse Problem ◽  
Linear Algebra ◽  
Matrix Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Explicit Solutions ◽  
Matrix Inverse ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Special Case

AbstractIn this article we consider Re-nnd solutions of the equation AXB=C with respect to X, where A,B,C are given matrices. We give necessary and sufficient conditions for the existence of Re-nnd solutions and present a general form of such solutions. As a special case when A=I we obtain the results from a paper of Groß (‘Explicit solutions to the matrix inverse problem AX=B’, Linear Algebra Appl.289 (1999), 131–134).

Matrix Formalism for Complex Heat Exchangers

1984 ◽  
Vol 106 (2) ◽  
pp. 352-360 ◽  
Author(s):  
A. Pignotti
Keyword(s):  
Finite Number ◽  
Model Calculation ◽  
Heat Exchangers ◽  
Practical Interest ◽  
Explicit Solutions ◽  
Matrix Formalism ◽  
Normal Flow ◽  
The Matrix ◽  
Shell And Tube

The matrix formalism introduced by Domingos for the calculation of the effectiveness of assemblies of heat exchangers is extended to the case of complex exchangers, for which the restriction of complete mixing for each fluid in the inlet and outlet streams is relaxed. Rectangular matrices are used to relate the inlet and outlet temperatures, and their properties are discussed. The method can be used to find explicit solutions for configurations of practical interest, which are analyzed as assemblies of simpler units, coupled together in various ways. The formalism is illustrated by a model calculation of a 1–2 shell-and-tube exchanger with a finite number of baffles, for TEMA E (normal flow) and TEMA J (divided flow) types of shells.

Evaluation of the matrix elements and for asymmetric rotors

1969 ◽  
Vol 47 (5) ◽  
pp. 509-511
Author(s):  
J. H. Wray
Keyword(s):  
Explicit Solutions ◽  
Rigid Rotor ◽  
Matrix Elements ◽  
The Matrix

A method of computing the matrix elements [Formula: see text] and [Formula: see text] for asymmetric rotors is presented which utilizes the rigid rotor reduced energies EτJ(κ). The method is formally the same as that of Kivelson and Wilson but differs in the form of the Hamiltonian considered. Explicit solutions of [Formula: see text] and [Formula: see text] are given for those levels for which EτJ(κ) may be given explicitly. These levels are those for which the roots of the reduced energy submatrices can be derived from linear or quadratic factors.

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