Some categories associated with bases of the kalman algebra

We investigate relationships in the Kalman algebra, viewed as an algebra over the algebra induced by the coefficients of the characteristic polynomial of the state matrix. To this end we introduce the categories BSS \mathcal{B}\mathcal{S}\mathcal{S} and BSS^\textstrict \mathcal{B}\mathcal{S}{\mathcal{S}^{\text{strict}}} associated with relationships in some basis of the Kalman algebra and also with reconstruction of relationships from known fragments. On these categories we construct the structures of the symmetrical monoidal category induced by addition and multiplication in the Kalman algebra. We investigate the properties of some of the most important classes of morphisms, in particular, we describe the structure and the action of the automorphism group.     

(Super) module amenability, module topological centre and semigroup algebras

For a Banach algebra $\mathcal{A}For a Banach algebra A\mathcal{A} which is also an \mathfrakA\mathfrak{A}-bimodule, we study relations between module amenability of closed subalgebras of A"\mathcal{A}', which contains A\mathcal{A}, and module Arens regularity of A\mathcal{A} and the role of the module topological centre in module amenability of A"\mathcal{A}'. Then we apply these results to A=l¹(S)\mathcal{A}=l^{1}(S) and \mathfrakA=l¹(E)\mathfrak{A}=l^{1}(E) for an inverse semigroup S with subsemigroup E of idempotents. We also show that l ¹(S) is module amenable (equivalently, S is amenable) if and only if an appropriate group homomorphic image of S, the discrete group \fracS ? \frac{S}{\approx}, is amenable. Moreover, we define super module amenability and show that l ¹(S) is super module amenable if and only if \fracS ? \frac{S}{\approx} is finite.     

The Asymptotics of Wilkinson’s Shift: Loss of Cubic Convergence

One of the most widely used methods for eigenvalue computation is the QR iteration with Wilkinson’s shift: Here, the shift s is the eigenvalue of the bottom 2×2 principal minor closest to the corner entry. It has been a long-standing question whether the rate of convergence of the algorithm is always cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let $T_{ {\mathcal {X}}}One of the most widely used methods for eigenvalue computation is the QR iteration with Wilkinson’s shift: Here, the shift s is the eigenvalue of the bottom 2×2 principal minor closest to the corner entry. It has been a long-standing question whether the rate of convergence of the algorithm is always cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let T _XT_{ {\mathcal {X}}} be the 3×3 matrix having only two nonzero entries (T _X)₁₂=(T _X)₂₁=1(T_{ {\mathcal {X}}})_{12}=(T_{ {\mathcal {X}}})_{21}=1 and let T_\varLambda {\mathcal {T}}_{\varLambda } be the set of real, symmetric tridiagonal matrices with the same spectrum as T _XT_{ {\mathcal {X}}} . There exists a neighborhood U ì T_\varLambda \boldsymbol {{\mathcal {U}}}\subset {\mathcal {T}}_{\varLambda } of T _XT_{ {\mathcal {X}}} which is invariant under Wilkinson’s shift strategy with the following properties. For T₀ ? UT_{0}\in \boldsymbol {{\mathcal {U}}} , the sequence of iterates (T _k ) exhibits either strictly quadratic or strictly cubic convergence to zero of the entry (T _k )₂₃. In fact, quadratic convergence occurs exactly when limT_k=T _X\lim T_{k}=T_{ {\mathcal {X}}} . Let X\boldsymbol {{\mathcal {X}}} be the union of such quadratically convergent sequences (T _k ): The set X\boldsymbol {{\mathcal {X}}} has Hausdorff dimension 1 and is a union of disjoint arcs X^s\boldsymbol {{\mathcal {X}}}^{\sigma} meeting at T _XT_{ {\mathcal {X}}} , where σ ranges over a Cantor set.     

Implicit QR algorithms for palindromic and even eigenvalue problems

In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version. Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence. We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques. D. S. Watkins partly supported by Deutsche Forschungsgemeinschaft through Matheon, the DFG Research Center Mathematics for key technologies in Berlin.     

Contraction and Expansion of Convex Sets

Let S{\mathcal{S}} be a set system of convex sets in ℝ ^d . Helly’s theorem states that if all sets in S{\mathcal{S}} have empty intersection, then there is a subset S￠ ì S{\mathcal{S}}'\subset{\mathcal{S}} of size d+1 which also has empty intersection. The conclusion fails, of course, if the sets in S{\mathcal{S}} are not convex or if S{\mathcal{S}} does not have empty intersection. Nevertheless, in this work we present Helly-type theorems relevant to these cases with the aid of a new pair of operations, affine-invariant contraction, and expansion of convex sets. These operations generalize the simple scaling of centrally symmetric sets. The operations are continuous, i.e., for small ε>0, the contraction C ^−ε and the expansion C ^ε are close (in the Hausdorff distance) to C. We obtain two results. The first extends Helly’s theorem to the case of set systems with nonempty intersection:     

On Linear Fractional Transformations Associated with Generalized J-Inner Matrix Functions

A class U_k1 (J){\mathcal{U}}_{\kappa 1} (J) of generalized J-inner mvf’s (matrix valued functions) W(λ) which appear as resolvent matrices for bitangential interpolation problems in the generalized Schur class of p ×q  mvf￠s S_k^{p ×q}p \times q \, {\rm mvf's}\, {\mathcal{S}}_{\kappa}^{p \times q} and some associated reproducing kernel Pontryagin spaces are studied. These spaces are used to describe the range of the linear fractional transformation T_W based on W and applied to S_k2^{p ×q}{\mathcal{S}}_{\kappa 2}^{p \times q}. Factorization formulas for mvf’s W in a subclass U^°_k1 (J) of U_k1(J){\mathcal{U}^{\circ}_{\kappa 1}} (J)\, {\rm of}\, {\mathcal{U}}_{\kappa 1}(J) found and then used to parametrize the set S_k1+k2^{p ×q} ?T_W [ S_k2^{p ×q} ]{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} \cap T_{W} \left[ {\mathcal{S}}_{\kappa 2}^{p \times q} \right]. Applications to bitangential interpolation problems in the class S_k1+k2^{p ×q}{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} will be presented elsewhere.     

Repeated and Continuous Interactions in Open Quantum Systems

We consider a finite quantum system S{\mathcal {S}} coupled to two environments of different nature. One is a heat reservoir R{\mathcal {R}} (continuous interaction) and the other one is a chain C{\mathcal {C}} of independent quantum systems E{\mathcal {E}} (repeated interaction). The interactions of S{\mathcal {S}} with R{\mathcal {R}} and C{\mathcal {C}} lead to two simultaneous dynamical processes. We show that for generic such systems, any initial state approaches an asymptotic state in the limit of large times. We express the latter in terms of the resonance data of a reduced propagator of S+R{\mathcal {S}+\mathcal {R}} and show that it satisfies a second law of thermodynamics. We analyze a model where both S{\mathcal {S}} and E{\mathcal {E}} are two-level systems and obtain the asymptotic state explicitly (at lowest order in the interaction strength). Even though R{\mathcal {R}} and C{\mathcal {C}} are not directly coupled, we show that they exchange energy, and we find the dependence of this exchange in terms of the thermodynamic parameters. We formulate the problem in the framework of W ^*-dynamical systems and base the analysis on a combination of spectral deformation methods and repeated interaction model techniques. We analyze the full system via rigorous perturbation theory in the coupling strength, and do not resort to any scaling limit, like e.g. weak coupling limits, or any other approximations in order to derive some master equation.     

Congruence-simple subsemirings of ℚ+

Commutative congruence-simple semirings have already been characterized with the exception of the subsemirings of ℝ⁺. Even the class CongSimp(\mathbb Q⁺)\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb {Q}^{+}) of all congruence-simple subsemirings of ℚ⁺ has not been classified yet. We introduce a new large class of the congruence-simple saturated subsemirings of ℚ⁺. We classify all the maximal elements of CongSimp(\mathbbQ⁺)\mathit{\mathcal{C}ong\mathcal {S}imp}(\mathbb{Q}^{+}) and show that every element of CongSimp(\mathbbQ⁺)\{\mathbbQ⁺}\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb{Q}^{+})\setminus\{\mathbb{Q}^{+}\} is contained in at least one of them.     

On approximation algorithms for concave mixed-integer quadratic programming

Concave mixed-integer quadratic programming is the problem of minimizing a concave quadratic polynomial over the mixed-integer points in a polyhedral region. In this work we describe an algorithm that finds an \(\epsilon \)-approximate solution to a concave mixed-integer quadratic programming problem. The running time of the proposed algorithm is polynomial in the size of the problem and in \(1/\epsilon \), provided that the number of integer variables and the number of negative eigenvalues of the objective function are fixed. The running time of the proposed algorithm is expected unless \(\mathcal {P}=\mathcal {NP}\).     

Separable standard quadratic optimization problems

A standard quadratic optimization problems (StQP) asks for the minimal value of a quadratic form over the standard simplex. StQPs form a central NP-hard class in quadratic optimization and have numerous practical applications. In this note we study the case of a separable objective function and propose an algorithm of worst-case complexity O(nlogn){\mathcal{O}(n\log n)} . Some extensions to multi-StQPs and ℓ ¹−ball constrained problems are also addressed briefly.     

Ergodic Subequivalence Relations Induced by a Bernoulli Action

Let Γ be a countable group and denote by S{\mathcal{S}} the equivalence relation induced by the Bernoulli action G\curvearrowright [0, 1]^G{\Gamma\curvearrowright [0, 1]^{\Gamma}}, where [0, 1]^Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation R{\mathcal{R}} of S{\mathcal{S}}, there exists a partition {X _i } _i≥0 of [0, 1]^Γ into R{\mathcal{R}}-invariant measurable sets such that R_|X0{\mathcal{R}_{\vert X_{0}}} is hyperfinite and R_|Xi{\mathcal{R}_{\vert X_{i}}} is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1.     

On invariant notions of Segre varieties in binary projective spaces

Invariant notions of a class of Segre varieties S_(m)(2){\mathcal{S}_{(m)}(2)} of PG(2 ^m − 1, 2) that are direct products of m copies of PG(1, 2), m being any positive integer, are established and studied. We first demonstrate that there exists a hyperbolic quadric that contains S_(m)(2){\mathcal{S}_{(m)}(2)} and is invariant under its projective stabiliser group G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} . By embedding PG(2 ^m − 1, 2) into PG(2 ^m − 1, 4), a basis of the latter space is constructed that is invariant under G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} as well. Such a basis can be split into two subsets whose spans are either real or complex-conjugate subspaces according as m is even or odd. In the latter case, these spans can, in addition, be viewed as indicator sets of a G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} -invariant geometric spread of lines of PG(2 ^m − 1, 2). This spread is also related with a G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} -invariant non-singular Hermitian variety. The case m = 3 is examined in detail to illustrate the theory. Here, the lines of the invariant spread are found to fall into four distinct orbits under G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} , while the points of PG(7, 2) form five orbits.     

Topology and Smooth Structure for Pseudoframes

Given a closed subspace ${\mathcal{S}}Given a closed subspace S{\mathcal{S}} of a Hilbert space H{\mathcal{H}}, we study the sets F_S{\mathcal{F}_\mathcal{S}} of pseudo-frames, CF_S{\mathcal{C}\mathcal{F}_\mathcal{S}} of commutative pseudo-frames and \mathfrakX_S{\tiny{\mathfrak{X}}_{\mathcal{S}}} of dual frames for S{\mathcal{S}}, via the (well known) one to one correspondence which assigns a pair of operators (F, H) to a frame pair ({f_n}_{n ? \mathbbN},{h_n}_{n ? \mathbbN}){(\{f_n\}_{n\in\mathbb{N}},\{h_n\}_{n\in\mathbb{N}})},

F:l2? H,     F({cn}n ? \mathbbN )=?n cn fn,F:\ell^2\to\,\mathcal{H}, \quad F\left(\{c_n\}_{n\in\mathbb{N}} \right)=\sum_n c_n f_n,      14. Nilpotent algebras and affinely homogeneous surfaces      Gregor?Fels  Wilhelm?KaupEmail author 《Mathematische Annalen》2012,353(4):1315-1350 To every nilpotent commutative algebra N{\mathcal{N}} of finite dimension over an arbitrary base field of characteristic zero a smooth algebraic subvariety S ì N{S\subset\mathcal{N}} can be associated in a canonical way whose degree is the nil-index and whose codimension is the dimension of the annihilator A{\mathcal{A}} of N{\mathcal{N}}. In case N{\mathcal{N}} admits a grading, the surface S is affinely homogeneous. More can be said if A{\mathcal{A}} has dimension 1, that is, if N{\mathcal{N}} is the maximal ideal of a Gorenstein algebra. In this case two such algebras N{\mathcal{N}}, [(N)\tilde]{\tilde{\mathcal{N}}} are isomorphic if and only if the associated hypersurfaces S, [(S)\tilde]{\tilde S} are affinely equivalent. If one of S, [(S)\tilde]{\tilde S} even is affinely homogeneous, ‘affinely equivalent’ can be replaced by ‘linearly equivalent’. In case the nil-index of N{\mathcal{N}} does not exceed 4 the hypersurface S is always affinely homogeneous. Contrary to the expectation, in case nil-index 5 there exists an example (in dimension 23) where S is not affinely homogeneous.      15. On the prime number theorem for the Selberg class      J. Kaczorowski  A. Perelli 《Archiv der Mathematik》2003,80(3):255-263 We show that the prime number theorem is equivalent with the non-vanishing on the 1-line, in the general setting of the Selberg class S \mathcal{S} of L \mathcal{L} -functions. The proof is based on a weak zero-density estimate near the 1-line and on a simple almost periodicity argument. We also give a conditional proof of the non-vanishing on the 1-line for every L \mathcal{L} -function in S \mathcal{S} , assuming a certain normality conjecture.      16. Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips      D. Danielli  N. Garofalo  D. M. Nhieu 《Mathematische Zeitschrift》2010,265(3):617-637 We consider the class of minimal surfaces given by the graphical strips ${{\mathcal S}}We consider the class of minimal surfaces given by the graphical strips S{{\mathcal S}} in the Heisenberg group \mathbb H1{{\mathbb {H}}^1} and we prove that for points p along the center of \mathbb H1{{\mathbb {H}}^1} the quantity \fracsH(S?B(p,r))rQ-1{\frac{\sigma_H(\mathcal S\cap B(p,r))}{r^{Q-1}}} is monotone increasing. Here, Q is the homogeneous dimension of \mathbb H1{{\mathbb {H}}^1} . We also prove that these minimal surfaces have maximum volume growth at infinity.      17. Inverse median location problems with variable coordinates      Fahimeh Baroughi Bonab  Rainer E. Burkard  Behrooz Alizadeh 《Central European Journal of Operations Research》2010,18(3):365-381 Given n points in \mathbbRd{\mathbb{R}^d} with nonnegative weights, the inverse 1-median problem with variable coordinates consists in changing the coordinates of the given points at minimum cost such that a prespecified point in \mathbbRd{\mathbb{R}^d} becomes the 1-median. The cost is proportional to the increase or decrease of the corresponding point coordinate. If the distances between points are measured by the rectilinear norm, the inverse 1-median problem is NP{\mathcal{NP}}-hard, but it can be solved in pseudo-polynomial time. Moreover, a fully polynomial time approximation scheme exists in this case. If the point weights are assumed to be equal, the corresponding inverse problem can be reduced to d continuous knapsack problems and is therefore solvable in O(nd) time. In case that the squared Euclidean norm is used, we derive another efficient combinatorial algorithm which solves the problem in O(nd) time. It is also shown that the inverse 1-median problem endowed with the Chebyshev norm in the plane is NP{\mathcal{NP}}-hard. Another pseudo-polynomial algorithm is developed for this case, but it is shown that no fully polynomial time approximation scheme does exist.      18. Second module cohomology group of inverse semigroup algebras      E. Nasrabadi  A. Pourabbas 《Semigroup Forum》2010,81(2):269-276 Let S be a commutative inverse semigroup and let E be its subsemigroup of idempotents. In this paper we define the n-th module cohomology group of Banach algebras and we show that H2l1(E)(l1(S),l1(S)(n))\mathcal {H}^{2}_{\ell^{1}(E)}(\ell^{1}(S),\ell^{1}(S)^{(n)}) is a Banach space for every odd n∈ℕ.      19. Deforming the Lie superalgebra of contact vector fields on <Emphasis Type="Italic">S</Emphasis><Superscript>1|2</Superscript> inside the Lie superalgebra of pseudodifferential operators on <Emphasis Type="Italic">S</Emphasis><Superscript>1|2</Superscript>      N. Ben Fraj  S. Omri 《Theoretical and Mathematical Physics》2010,163(2):618-633 We classify deformations of the standard embedding of the Lie superalgebra $ \mathcal{K} $ \mathcal{K} (2) of contact vector fields on the (1, 2)-dimensional supercircle into the Lie superalgebra SΨD(S 1|2 ) of pseudodifferential operators on the supercircle S 1|2 . The proposed approach leads to the deformations of the central charge induced on $ \mathcal{K} $ \mathcal{K} (2) by the canonical central extension of SΨD(S 1|2 ).      20. An inexact primal–dual path following algorithm for convex quadratic SDP      Kim-Chuan Toh 《Mathematical Programming》2008,112(1):221-254 We propose primal–dual path-following Mehrotra-type predictor–corrector methods for solving convex quadratic semidefinite programming (QSDP) problems of the form: , where is a self-adjoint positive semidefinite linear operator on , b∈R m , and is a linear map from to R m . At each interior-point iteration, the search direction is computed from a dense symmetric indefinite linear system (called the augmented equation) of dimension m + n(n + 1)/2. Such linear systems are typically very large and can only be solved by iterative methods. We propose three classes of preconditioners for the augmented equation, and show that the corresponding preconditioned matrices have favorable asymptotic eigenvalue distributions for fast convergence under suitable nondegeneracy assumptions. Numerical experiments on a variety of QSDPs with n up to 1600 are performed and the computational results show that our methods are efficient and robust. Research supported in part by Academic Research Grant R146-000-076-112.

Nilpotent algebras and affinely homogeneous surfaces

To every nilpotent commutative algebra N{\mathcal{N}} of finite dimension over an arbitrary base field of characteristic zero a smooth algebraic subvariety S ì N{S\subset\mathcal{N}} can be associated in a canonical way whose degree is the nil-index and whose codimension is the dimension of the annihilator A{\mathcal{A}} of N{\mathcal{N}}. In case N{\mathcal{N}} admits a grading, the surface S is affinely homogeneous. More can be said if A{\mathcal{A}} has dimension 1, that is, if N{\mathcal{N}} is the maximal ideal of a Gorenstein algebra. In this case two such algebras N{\mathcal{N}}, [(N)\tilde]{\tilde{\mathcal{N}}} are isomorphic if and only if the associated hypersurfaces S, [(S)\tilde]{\tilde S} are affinely equivalent. If one of S, [(S)\tilde]{\tilde S} even is affinely homogeneous, ‘affinely equivalent’ can be replaced by ‘linearly equivalent’. In case the nil-index of N{\mathcal{N}} does not exceed 4 the hypersurface S is always affinely homogeneous. Contrary to the expectation, in case nil-index 5 there exists an example (in dimension 23) where S is not affinely homogeneous.     

On the prime number theorem for the Selberg class

We show that the prime number theorem is equivalent with the non-vanishing on the 1-line, in the general setting of the Selberg class S \mathcal{S} of L \mathcal{L} -functions. The proof is based on a weak zero-density estimate near the 1-line and on a simple almost periodicity argument. We also give a conditional proof of the non-vanishing on the 1-line for every L \mathcal{L} -function in S \mathcal{S} , assuming a certain normality conjecture.     

Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips

We consider the class of minimal surfaces given by the graphical strips ${{\mathcal S}}We consider the class of minimal surfaces given by the graphical strips S{{\mathcal S}} in the Heisenberg group \mathbb H¹{{\mathbb {H}}^1} and we prove that for points p along the center of \mathbb H¹{{\mathbb {H}}^1} the quantity \fracs_H(S?B(p,r))r^Q-1{\frac{\sigma_H(\mathcal S\cap B(p,r))}{r^{Q-1}}} is monotone increasing. Here, Q is the homogeneous dimension of \mathbb H¹{{\mathbb {H}}^1} . We also prove that these minimal surfaces have maximum volume growth at infinity.     

Inverse median location problems with variable coordinates

Given n points in \mathbbR^d{\mathbb{R}^d} with nonnegative weights, the inverse 1-median problem with variable coordinates consists in changing the coordinates of the given points at minimum cost such that a prespecified point in \mathbbR^d{\mathbb{R}^d} becomes the 1-median. The cost is proportional to the increase or decrease of the corresponding point coordinate. If the distances between points are measured by the rectilinear norm, the inverse 1-median problem is NP{\mathcal{NP}}-hard, but it can be solved in pseudo-polynomial time. Moreover, a fully polynomial time approximation scheme exists in this case. If the point weights are assumed to be equal, the corresponding inverse problem can be reduced to d continuous knapsack problems and is therefore solvable in O(nd) time. In case that the squared Euclidean norm is used, we derive another efficient combinatorial algorithm which solves the problem in O(nd) time. It is also shown that the inverse 1-median problem endowed with the Chebyshev norm in the plane is NP{\mathcal{NP}}-hard. Another pseudo-polynomial algorithm is developed for this case, but it is shown that no fully polynomial time approximation scheme does exist.     

Second module cohomology group of inverse semigroup algebras

Let S be a commutative inverse semigroup and let E be its subsemigroup of idempotents. In this paper we define the n-th module cohomology group of Banach algebras and we show that H²_l¹(E)(l¹(S),l¹(S)⁽ⁿ⁾)\mathcal {H}^{2}_{\ell^{1}(E)}(\ell^{1}(S),\ell^{1}(S)^{(n)}) is a Banach space for every odd n∈ℕ.     

Deforming the Lie superalgebra of contact vector fields on <Emphasis Type="Italic">S</Emphasis><Superscript>1|2</Superscript> inside the Lie superalgebra of pseudodifferential operators on <Emphasis Type="Italic">S</Emphasis><Superscript>1|2</Superscript>

We classify deformations of the standard embedding of the Lie superalgebra $ \mathcal{K} $ \mathcal{K} (2) of contact vector fields on the (1, 2)-dimensional supercircle into the Lie superalgebra SΨD(S ^1|2 ) of pseudodifferential operators on the supercircle S ^1|2 . The proposed approach leads to the deformations of the central charge induced on $ \mathcal{K} $ \mathcal{K} (2) by the canonical central extension of SΨD(S ^1|2 ).     

An inexact primal–dual path following algorithm for convex quadratic SDP

We propose primal–dual path-following Mehrotra-type predictor–corrector methods for solving convex quadratic semidefinite programming (QSDP) problems of the form: , where is a self-adjoint positive semidefinite linear operator on , b∈R ^m , and is a linear map from to R ^m . At each interior-point iteration, the search direction is computed from a dense symmetric indefinite linear system (called the augmented equation) of dimension m + n(n + 1)/2. Such linear systems are typically very large and can only be solved by iterative methods. We propose three classes of preconditioners for the augmented equation, and show that the corresponding preconditioned matrices have favorable asymptotic eigenvalue distributions for fast convergence under suitable nondegeneracy assumptions. Numerical experiments on a variety of QSDPs with n up to 1600 are performed and the computational results show that our methods are efficient and robust. Research supported in part by Academic Research Grant R146-000-076-112.