Semidefinite Research Articles

Sharing the care of mechanical circulatory support: collaborative efforts of patients/caregivers, shared-care sites, and left ventricular assist device implanting centers.

Left ventricular assist devices (LVADs) improve longevity, as well as functional capacity and quality of life in patients with medically refractory heart failure.1–5 During the 6 years since the US Food and Drug Administration approval of the first commercially available continuous flow LVAD in 2008, the number of patients supported by these devices has grown exponentially. The most recent report from the Interagency Registry for Mechanically Assisted Circulatory Support recorded nearly 2500 North American LVAD implants in 2013 at >150 implanting centers.6 Approximately 40% of LVADs are implanted as an indefinite form of support (ie, destination therapy [DT]), and nearly half of patients are surviving >4 years.6 This combination of improved survival and evolving implant strategies has led to an increasing number of LVAD recipients being integrated into the community and we are observing the growth of an ambulatory LVAD population with a unique set of clinical needs. There are 2 predominant strategies of LVAD implantation. Patients who are not eligible for transplantation because of issues such as advanced age may be considered for an LVAD as DT, implying that they will live the rest of their lives with the LVAD. This is in contrast to patients who undergo LVAD implantation as a temporary support device, to support them while awaiting transplantation. This later strategy is defined as a bridge-to-transplantation, following which, the device will be explanted. Centers implanting LVADs without concomitant transplant programs (designated DT centers) have emerged to provide this therapy, averting the need for travel to more distant transplant centers. Such DT centers operate in conjunction with a VAD/transplant partner in the care of patients awaiting transplantation. Because LVAD implant centers are frequently remotely located from the patients that they serve, a model of shared-care has developed, whereby the continued care of …

On a question of Uri Shapira and Barak Weiss

Here it is proved that if Q(x1,..., xn) is a positive definite quadratic form which is reduced in the sense of Korkine and Zolotareff and has outer coefficients B1,..., Bn satisfying B1 ≥ 1) Bn ≤ 1 and B1 ⋯ Bn = 1, then its inhomogeneous minimum is at most n/4 for n ≤ 7. This result implies a positive answer to a question of Shapira and Weiss for stable lattices and thereby provides another proof of Minkowski’s Conjecture on the product of n real non-homogeneous linear forms in n variables for n ≤ 7. Our result is an analogue of Woods’ Conjecture which has been proved for n ≤ 9. The analogous problem when B11 is also investigated.

Some congruences for Siegel theta series

We discuss an arithmetic approach to some congruence properties of Siegel theta series of even positive definite unimodular quadratic forms.

Bound for the fifth coefficient of certain starlike functions

For two different choices of φ, the sharp bound for the fifth coefficient of a normalized analytic function f satisfying zf′(z)/f(z)≺φ(z) is obtained by using a bound for a polynomial in the coefficients of functions with positive real part. Our proof uses a characterization of functions with positive real part in terms of certain positive semi-definite Hermitian form and certain well-known coefficient inequalities for functions with positive real part are shown to follow easily from this characterization.

Erratum to: On odd rank integral quadratic forms: canonical representatives of projective classes and explicit construction of integral classes with square-free determinant

Two rank \(n\), integral quadratic forms \(f\) and \(g\) are said projectively equivalent if there exist nonzero rational numbers \(r\) and \(s\) such that \(rf\) and \(sg\) are rationally equivalent. Two odd dimensional, integral quadratic forms \(f\) and \(g\) are projectivelly equivalent if and only if their adjoints are rationally equivalent. We prove that a canonical representative of each projective class of forms of odd rank, exists and is unique up to genus (integral equivalence for indefinite forms). We give a useful characterization of this canonical representative. An explicit construction of integral classes with square-free determinant is given. As a consequence, two tables of ternary and quinary integral quadratic forms of index \(1\) and with square-free determinant are presented.

Signal Design for Context Aware Distributed Radar Sensing Networks Based on Wavelets

In the field of radar sensing, a paradigm shift is taking place from the traditional radar system to distributed sensing/radar networks. Comparing with the single radar system distributed radar networks have a superior performance in detection and estimation of parameters of target. They also enhance the imaging and classification performance of sensing systems. As in distributed radar networks several radars operate at the same frequency band in a network and share the spectrum they may interfere each other, and accordingly the radar-to-radar interference becomes a major problem to be addressed. Therefore, the context aware design of radar signals for radar sensing networks to cope with the interference is pronounced. On the other hand Wavelet packet transform has recently emerged as a novel signal design technique with attractions such as good time–frequency resolution, low sidelobes, and the reconfigurability capability. The wavelet approach is advantageous for the signal design of cognitive radar networks mainly because of its flexibility, lower sensitivity to distortion and interference, and better utilization of spectrum. For minimum interference to the adjacent radar bands, the wavelets should be maximally frequency selective. Commonly known wavelets are not frequency selective in nature and hence result in poor spectrum sharing performance. To alleviate this problem, we design a family of wavelets that are maximally frequency selective in nature. To this end, the design constraints are first enlisted. Then the problem, originally non-convex, is reformulated into a convex optimization problem and solved using Semi Definite Programming tools. Through simulation studies the benefits of the newly designed wavelets for a distributed radar sensing network are demonstrated.

Rational period functions and cycle integrals in higher level cases

Generalizing the results of Duke, Imamoḡlu, and Tóth [8] we give an effective basis for the space of period polynomials in higher level cases and construct modular integrals for certain rational period functions related to indefinite binary quadratic forms by means of cycle integrals.

S-Duality for surfaces with $$A_n$$ A n -type singularities

We show that the generating series of Euler characteristics of Hilbert schemes of points on any algebraic surface with at worst \(A_n\)-type singularities is described by the theta series determined by integer valued positive definite quadratic forms and the Dedekind eta function. In particular it is a Fourier development of a meromorphic modular form with possibly half integer weight. The key ingredient is to apply the flop transformation formula of Donaldson–Thomas type invariants counting two dimensional torsion sheaves on threefolds proved in the author’s previous paper.

The Laplacian quadratic form and edge connectivity of a graph

Let G be a simple connected graph with associated positive semidefinite integral quadratic form Q(x) = \sum (x(i) − x(j))^2, where the sum is taken over all edges ij of G. It is showed that the minimum positive value of Q(x) for x ∈ Z_n equals the edge connectivity of G. By restricting Q(x) to x ∈ Z_{n−1} × {0}, the quadratic form becomes positive definite. It is also showed that the number of minimal disconnecting sets of edges of G equals twice the number of vectors x ∈ Z_{n−1} ×{0} for which the form Q attains its minimum positive value.

A generalized Hermite constant for imaginary quadratic fields

We introduce the projective Hermite constant for positive definite binary hermitian forms associated with an imaginary quadratic number field $K$. It is a lower bound for the classical Hermite constant, and these two constants coincide when $K$ has class number one. Using the geometric tools developed by Mendoza and Vogtmann for their study of the homology of the Bianchi groups, we compute the projective Hermite constants for those $K$ whose absolute discriminants are less than 70, and determine the hermitian forms that attain the projective Hermite constants in these cases. A comparison of the projective hermitian constant with some other generalizations of the classical Hermite constant is also given.

On the Real Zeros of Positive Semidefinite Biquadratic Forms

For a positive semidefinite biquadratic forms F in (3, 3) variables, we prove that, if F has a finite number but at least 7 real zeros 𝒵(F), then it is not a sum of squares. We show also that if F has at least 11 zeros, then it has infinitely many real zeros and it is a sum of squares. It can be seen as the counterpart for biquadratic forms as the results of Choi, Lam, and Reznick for positive semidefinite ternary sextics. We introduce and compute some of the numbers BB n, m which are set to be equal to sup |𝒵(F)| where F ranges over all the positive semidefinite biquadratic forms F in (n, m) variables such that |𝒵(F)| < ∞. We also recall some old constructions of positive semidefinite biquadratic forms which are not sums of squares and we give some new families of examples.

Finite-Horizon Distributed H∞ Fault Estimation for Time-Varying Systems in Sensor Networks: A Krein-Space Approach

In this paper, the finite-horizon distributed H∞ fault estimation problem is investigated for a class of linear discrete time-varying systems in sensor networks. Sensor nodes are distributed according to a given network topology and each sensor receives the information from both the system and its neighbors. A set of measurement models is established for each sensor node that describes all the available information from the node and its neighbors. The distributed H∞ fault estimation problem addressed is first transformed into an optimization issue of a certain indefinite quadratic form. Then, by employing innovation analysis and projection theory in Krein spaces, the desired distributed H∞ fault estimators are designed in terms of the solution to a set of recursive equations. Finally, an illustrative example is presented to show the effectiveness of the proposed distributed fault estimation scheme.

Using Lyapunov Functions to Construct Lyapunov Functionals for Delay Differential Equations

Given that a Lyapunov function is known for a particular system, we outline an approach for determining terms in the system that can be replaced by similar terms that include delay, without changing the global stability. The approach is based on adding integral terms to the original Lyapunov function so that the new Lyapunov derivative is still negative semidefinite. This gives a Lyapunov functional for the modified system. The delay structures that are permitted include discrete delay, distributed delay, and combinations of the two. Several examples involving compartmental models in virology and epidemiology are included to show the effectiveness of the method, as well as the ease with which it can be applied.

Theta series associated with certain positive definite binary quadratic forms

Recently, P. C. Toh derived identities expressing theta series associated with fundamental idoneal discriminants in terms of generalized Lambert series. His method depends on the fact that the ring of integers of a number field is a Dedekind domain, and i

On values of binary quadratic forms at integer points

We obtain estimates for the number of integral solutions in large balls, of inequalities of the form $|Q(x, y)| < \epsilon$, where $Q$ is an indefinite binary quadratic form, in terms of the Hurwitz continued fraction expansions of the slopes of the lines on which $Q$ vanishes. The method is based on a coding of geodesics on the modular surface via Hurwitz expansions of the endpoints of their lifts in the Poincare half-plane.

H∞Fault Detection for Linear Discrete Time-Varying Descriptor Systems with Missing Measurements

This paper deals with the problem ofH∞fault detection for a class of linear discrete time-varying descriptor systems with missing measurements, and the missing measurements are described by a Bernoulli random binary switching sequence. We first translate theH∞fault detection problem into an indefinite quadratic form problem. Then, a sufficient and necessary condition on the existence of the minimum is derived. Finally, an observer-basedH∞fault detection filter is obtained such that the minimum is positive and its parameter matrices are calculated recursively by solving a matrix differential equation. A numerical example is given to demonstrate the efficiency of the proposed method.

A Semi Definite Programming Based Optimization Algorithm for Project Scheduling Problem

A method to estimate the time of accomplishment of a project is proposed in this study. The method claims a tight and better time complexity. A semi definite optimization based model for better calculation of project completion time in a project scheduling environment has been developed that works well when limited information of individual activities in the project is available. The computational experiments suggest that the result provided by the new method is more accurate. A significant improvement is achieved in terms of time complexities of algorithm compared to alternative methods.

Disaggregated econometric estimation of consumer demand response by alcoholic beverage types

The paper presents estimates of price elasticities of demand for 12 disaggregated alcoholic beverages in Australia: premium beer, full strength beer, low alcohol beer and mid-strength beer; red bottled wine, white bottled wine, sparkling wine, cask wine; dark and light ready-to-drink (RTD); and dark and light spirits. These disaggregated categories correspond closely to the commodities of interest to public policymakers with respect to taxation and health policies. The system of demand equations is estimated with Nielsen's data using a semiflexible Almost Ideal Demand System model in order to impose negative semi-definiteness on the demand parameters. Results indicate elastic own-price elasticities for virtually all commodities. Cross-price elasticities suggest that beverages most linked with negative externalities, namely full strength beer, dark RTD and dark spirits, may need to be taxed jointly. Any proposed tax increase to cask wine may also result in consumers shifting demand to more undesirable beverages. The elasticity estimates are used to illustrate the effect of a hypothetical change towards taxation equalisation based on alcohol content. These elasticities offer crucially needed inputs for analysing any tax change policies.

On quadratic Diophantine equations in four variables and orders associated with lattices II

The weighted average of the numbers of primitive solutions of a quadratic Diophantine equation in four variables connects with the mass of the special orthogonal group of a ternary quadratic form relative to a certain open subgroup, through the mass formula of Shimura. With the determination of suitable group indices, the computation of such a mass can be reduced to that of the mass of the genus of maximal lattices with respect to the ternary form. We determine those indices under several assumptions and provide the numerical examples of the weighted averages for a few positive-definite quaternary quadratic forms.

Ternary quadratic forms over number fields with small class number

We enumerate all positive definite ternary quadratic forms over number fields with class number at most 2. This is done by constructing all definite quaternion orders of type number at most 2 over number fields. Finally, we list all definite quaternion orders of ideal class number 1 or 2.