Determinant Det Research Articles

Strict convergence and minimal liftings in BV

Given a function υ ∈ BV (Ω; Rm), we introduce the notion of a minimal lifting of Dυ. We prove that every υ ∈ BV (Ω; Rm) has a unique minimal lifting, and we show that if υk → υ strictly in BV, then the minimal liftings of υk converge weakly as measures to the minimal lifting of υ. As an application, we deduce a result about weak continuity of the distributional determinant Det D2u with respect to strict convergence.

A Conceptual Proof of Cramer's Rule

Proof. The classical way to solve a linear equation system is by performing row operations: (i) add one row to another row, (ii) multiply a row with a nonzero scalar and (iii) exchange two rows. We show that the quotient in equation (1) will not change under row operations. Under the first row operation, the values of the two determinants det(Ai) and det(A) will not change, since determinants are invariant under this row operation. Under the second row operation both determinants will gain the same factor, which cancels in the quotient. Finally, under the third row operation both determinants will switch sign, which again cancels in the quotient. Since every invertible matrix A can be row reduced to the identity matrix, it is now enough to prove Cramer's rule for the identity matrix. However, this is a straightforward task. i

A Parent of Binet's Formula?

Proof. The classical way to solve a linear equation system is by performing row operations: (i) add one row to another row, (ii) multiply a row with a nonzero scalar and (iii) exchange two rows. We show that the quotient in equation (1) will not change under row operations. Under the first row operation, the values of the two determinants det(Ai) and det(A) will not change, since determinants are invariant under this row operation. Under the second row operation both determinants will gain the same factor, which cancels in the quotient. Finally, under the third row operation both determinants will switch sign, which again cancels in the quotient. Since every invertible matrix A can be row reduced to the identity matrix, it is now enough to prove Cramer's rule for the identity matrix. However, this is a straightforward task. i

Transfer matrices for the partition function of the Potts model on cyclic and Möbius lattice strips

We present a method for calculating transfer matrices for the q -state Potts model partition functions Z ( G , q , v ) , for arbitrary q and temperature variable v , on cyclic and Möbius strip graphs G of the square (sq), triangular (tri), and honeycomb (hc) lattices of width L y vertices and of arbitrarily great length L x vertices. For the cyclic case we express the partition function as Z ( Λ , L y × L x , q , v ) = ∑ d = 0 L y c ( d ) Tr [ ( T Z , Λ , L y , d ) m ] , where Λ denotes lattice type, c ( d ) are specified polynomials of degree d in q, T Z , Λ , L y , d is the transfer matrix in the degree- d subspace, and m = L x ( L x / 2 ) for Λ = sq , tri ( hc), respectively. An analogous formula is given for Möbius strips. We exhibit a method for calculating T Z , Λ , L y , d for arbitrary L y . Explicit results for arbitrary L y are given for T Z , Λ , L y , d with d = L y and L y - 1 . In particular, we find very simple formulas the determinant det ( T Z , Λ , L y , d ) , and trace Tr ( T Z , Λ , L y ) . Corresponding results are given for the equivalent Tutte polynomials for these lattice strips and illustrative examples are included. We also present formulas for self-dual cyclic strips of the square lattice.

Block TERM factorization of block matrices

Reversible integer mapping (or integer transform) is a useful way to realize lossless coding, and this technique has been used for multi-component image compression in the new international image compression standard JPEG 2000. For any nonsingular linear transform of finite dimension, its integer transform can be implemented by factorizing the transform matrix into 3 triangular elementary reversible matrices (TERMs) or a series of single-row elementary reversible matrices (SERMs). To speed up and parallelize integer transforms, we study block TERM and SERM factorizations in this paper. First, to guarantee flexible scaling manners, the classical determinant (det) is generalized to a matrix function,DET, which is shown to have many important properties analogous to those ofdet. Then based onDET, a generic block TERM factorization,BLUS, is presented for any nonsingular block matrix. Our conclusions can cover the early optimal point factorizations and provide an efficient way to implement integer transforms for large matrices.

Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions

We consider specific quantum mechanical model problems for which perturbation theory fails to explain physical properties like the eigenvalue spectrum even qualitatively, even if the asymptotic perturbation series is augmented by resummation prescriptions to “cure” the divergence in large orders of perturbation theory. Generalizations of perturbation theory are necessary, which include instanton configurations, characterized by non-analytic factors exp(− a/ g) where a is a constant and g is the coupling. In the case of one-dimensional quantum mechanical potentials with two or more degenerate minima, the energy levels may be represented as an infinite sum of terms each of which involves a certain power of a non-analytic factor and represents itself an infinite divergent series. We attempt to provide a unified representation of related derivations previously found scattered in the literature. For the considered quantum mechanical problems, we discuss the derivation of the instanton contributions from a semi-classical calculation of the corresponding partition function in the path integral formalism. We also explain the relation with the corresponding WKB expansion of the solutions of the Schrödinger equation, or alternatively of the Fredholm determinant det( H− E) (and some explicit calculations that verify this correspondence). We finally recall how these conjectures naturally emerge from a leading-order summation of multi-instanton contributions to the path integral representation of the partition function.

Block TERM factorization of block matrices

Reversible integer mapping (or integer transform) is a useful way to realize lossless coding, and this technique has been used for multi-component image compression in the new international image compression standard JPEG 2000. For any nonsingular linear transform of finite dimension, its integer transform can be implemented by factorizing the transform matrix into 3 triangular elementary reversible matrices (TERMs) or a series of single-row elementary reversible matrices (SERMs). To speed up and parallelize integer transforms, we study block TERM and SERM factorizations in this paper. First, to guarantee flexible scaling manners, the classical determinant (det) is generalized to a matrix function, DET , which is shown to have many important properties analogous to those of det . Then based on DET , a generic block TERM factorization, BLUS , is presented for any nonsingular block matrix. Our conclusions can cover the early optimal point factorizations and provide an efficient way to implement integer transforms for large matrices.

On the total variation of the Jacobian

A characterization of the total variation TV(u,Ω) of the Jacobian determinant det Du is obtained for some classes of functions u:Ω⊂ R 2→ R 2 outside the traditional regularity space W 1,2(Ω; R 2) . In particular, explicit formulas are deduced for functions that are locally Lipschitz continuous away from a given one point singularity x 0∈ Ω , i.e., u∈W 1,p(Ω; R 2)∩W 1,∞(Ω ⧹{x 0}; R 2) for some p>1.

A Non-Automatic (!) Application of Gosper's Algorithm Evaluates a Determinant from Tiling Enumeration

We evaluate the determinant det 1≤i,j≤n x (() - ()), which gives the number of lozenge tilings of a hexagon with cut off corners. A particularly interesting feature of this evaluation is that it requires the proof of a certain hypergeometric identity which we accomplish by using Gosper's algorithm in a nonautomatic fashion.

Spectre de l'opérateur de transfert en dimension 1

We study spectral properties of a transfer operator ℳΦ(x)=∑ωgω(x)Φ(ψωx) acting on functions of bounded variation. Using a symmetrical integral, we first obtain bounds on its spectral and essential spectral radii. We then consider the dynamical determinant Det#(Id +zℳ). Our main theorem generalizes to discontinuous weights the result of Baladi and Ruelle (for continuous weights) on the link between zeroes of the sharp determinant and eigenvalues of the transfer operator. The proof is based on regularizing the weights and uses a (new) spectral result giving the surjectivity of some applications between eigenspaces of operators.

Critical Metrics for the Determinant of the Laplacian in Odd Dimensions

Let M be a closed compact n-dimensional manifold with n odd. We calculate the first and second variations of the zeta-regularized determinants det'A and det L as the metric on M varies, where A denotes the Laplacian on functions and L denotes the conformal Laplacian. We see that the behavior of these functionals depends on the dimension. Indeed, every critical metric for (-1)(n-1)/2det'A or (_ 1)(n-1)/21 det L has finite index. Consequently there are no local maxima if n = 4rm + 1 and no local minima if n = 4m + 3. We show that the standard 3-sphere is a local maximum for det'A while the standard (4m + 3)-sphere with m = 1, 2,..., is a saddle point. By contrast, for all odd n, the standard n-sphere is a local extremal for det L. An important tool in our work is the canonical trace on odd class operators in odd dimensions. This trace is related to the determinant by the formula det Q = TR log Q, and we prove some basic results on how to calculate this trace.

Bloch electron in a magnetic field and the ising model

The spectral determinant det(H-varepsilonI) of the Azbel-Hofstadter Hamiltonian H is related to Onsager's partition function of the 2D Ising model for any value of magnetic flux Phi = 2piP/Q through an elementary cell, where P and Q are coprime integers. The band edges of H correspond to the critical temperature of the Ising model; the spectral determinant at these (and other points defined in a certain similar way) is independent of P. A connection of the mean of Lyapunov exponents to the asymptotic (large Q) bandwidth is indicated.

Calculation of a Certain Determinant

The n×n determinant det[(a+j−i)Γ(b+j+i)] is evaluated. This completes the calculation of the Mellin transform of the probability density of the determinant of a random quaternion self-dual matrix taken from the gaussian symplectic ensemble. The inverse Mellin transform then gives the later probability density itself.

The Vector Measures Whose Range Is Strictly Convex

Let μ be a measure on a measure space (X,Λ) with values in Rnandfbe the density of μ with respect to its total variation. We show that the range R(μ)={μ(E):E∈Λ} of μ is strictly convex if and only if the determinant det[f(x1),…,f(xn)] is nonzero a.e. onXn. We apply the result to a class of measures containing those that are generated by Chebyshev systems.

The Marcus-de Oliveira conjecture, bilinear forms, and cones

The well-known determinantal cinjecture of de Oliveira and Marcus (OMC) confines the determinant det ( X + Y) of the sum of normal n × n matrices X, Y to a certain region in the complex plane. Even the subconjecture obtained by specializing it to n = 4, X Hermitian and Y normal is still open. We view the subconjecture as a special case of an assertion concerning a certain family of bilinear forms of R 16 ×C 16 and give a method that may prove useful for establishing it for many of such matrix pairs, independent of their spectrum; in particular we apply it successfully in the case of a prominent unitary similarity of Drury's threatening OMC. Unfortunately we find the assertion, extended naturally to pairs of complex arguments to be false and the ideas outlined inapplicable for the general OMC( n=4) case. We also report on some computer experiments, formulate OMC( n=4) as a statement about cones, and find it would be implied by establishing the emptiness of certain semialgebraic sets defined by systems of quadratic and linear relations.

Norms and determinants of quaternionic line bundles

We investigate the relationship between the determinants det (P) and norms N ( P ) of right modules P of rank one over associative composition algebras C. We show: \( {\rm det} (P) \cong N(P) \otimes N(P) \) if C is a quaternion algebra, and \( {\rm det}\,(P) \cong N(P) \otimes {\rm det}\, (C)\) if C is a quadratic etale algebra.

Computing Eigenvalues and Eigenvectors Without Determinants

(1998). Computing Eigenvalues and Eigenvectors Without Determinants. Mathematics Magazine: Vol. 71, No. 1, pp. 24-33.

A Factorization of Determinant Related to Some Random Matrices

We consider the expectation of the determinant det(λ−X)−1for Im λ>0 associated with some random N×Nmatrices and factorize it into NStieltjes transforms of probability measures. Moreover, using this factorization, we investigate the limiting behavior of the logarithm of the quantity as N→∞.

Analysis of rotational invariants of the magnetotelluric impedance tensor

The rotational invariants of the magnetotelluric impedance tensor Z may serve as the most compact 3-D interpretational parameters, since they do not depend on the direction of the inducing field, and they may have various morphological characteristics over 3-D bodies. Their complete system is reviewed for the first time in this paper. It is demonstrated that the complex Z—having eight real-valued independent elements—has seven independent rotational invariants. The complex determinant det(Z) contains three independent real-valued invariants: det(ℛℯZ)–det(ImZ) and Imdet(Z) [where det(ReZ)–det(ImZ) = Re det(Z)] and not two, as is usually assumed from its complex character. The same is true for the sum of the squares of the elements of Z, ssq(Z) = Z2xx+ Z2xy+ Z2yx+ Z2yy. Its real-valued invariants are ssq ReZ) = Re2Zxx+ Re2Zxy+ Re2Zyx+ Re2Zyy; ssq(ImZ) = Im2Zxx+ Im2Zxy+ Im2Zyx+ Im2Zyy; and Im ssq(Z) = 2(ReZxx ImZxx+ ReZxy ImZxy+ ReZyx ImZyx+ ReZyy ImZyy) where Re ssq(Z) = ssq(ReZ)–ssq(ImZ), and ssq(ReZ) + ssq(ImZ) = ||Z||2f; here ||Z||f is the Frobenius norm of Z. The sets of seven independent rotational invariants can be selected in many different ways. In the classical magnetotelluric set, ReZ1, ImZ1 [where Z1 = (Zxy–Zyx)/2], ReZ2, ImZ2 (where the trace Zxx+ Zyy = 2Z2) and the three determinant-based real-valued invariants, det(ReZ)–det(ImZ) and Im det(Z), are suggested for use. If the trace, the determinant and ssq(Z) are accepted as basic scalar functions (we call them the mathematical selection of invariants), eight different sets of independent invariants can be selected. The geometrical meaning of rotational invariants is illustrated using two different graphic representations: complex-plane ellipses and Mohr circles. For electromagnetic imaging purposes it is suggested that some of those parameters that are derived from the real tensor ReZ should be used, since at realistic periods the model geometry of thin-sheet-like 3-D models is much better reflected in, for example, ReZ1, det(ReZ) and ssq(ReZ) than in any other invariants.

The Functional Determinant of a Four-Dimensional Boundary Value Problem

Working on four-dimensional manifolds with boundary, we consider, elliptic boundary value problems (A, B), A being the interior and B the boundary operator. These problems (A, B) should be valued in a tensorspinor bundle; should depend in a universal way on a Riemannian metric g and be formally selfadjoint; should behave in an appropriate way under conformal change $g \to {\Omega ^2}g$, $\Omega$ a smooth positive function; and the leading symbol of A should be positive definite. We view the functional determinant det ${A_B}$ of such a problem as a functional on a conformal class $\{ {\Omega ^2}g\}$, and develop a formula for the quotient of the determinant at ${\Omega ^2}g$ by that at g. (Analogous formulas are known to be intimately related to physical string theories in dimension two, and to sharp inequalities of borderline Sobolev embedding and Moser-Trudinger types for the boundariless case in even dimensions.) When the determinant in a background metric ${g_0}$ is explicitly computable, the result is a formula for the determinant at each metric ${\Omega ^2}{g_0}$ (not Just a quotient of determinants). For example, we compute the functional determinants of the Dirichlet and Robin (conformally covariant Neumann) problems for the Laplacian in the ball ${B^4}$, using our general quotient formulas in the case of the conformal Laplacian, together with an explicit computation on the hemisphere ${H^4}$.