Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings

A classical problem in real geometry concerns the representation of positive semidefinite elements of a ring A as sums of squares of elements of A. If A is an excellent ring of dimension $$\ge 3$$ ≥ 3 , it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in A. The one dimensional local case has been afforded by Scheiderer (mainly when its residue field is real closed). In this work we focus on the 2-dimensional case and determine (under some mild conditions) which local excellent henselian rings A of embedding dimension 3 have the property that every positive semidefinite element of A is a sum of squares of elements of A.

Stability Analysis of Rice Hybrids for Grain Yield in Telangana through AMMI and GGE Bi-plot Model

An investigation was carried out on fifteen rice genotypes to identify stable rice hybrids across six different agroclimatic zones in Telangana state using AMMI and GGE bi-plot analyses during July to November, 2020. Analysis of variance clearly showed that environments contributed highest (65.47%) in total sum of squares followed by genotypes×environments (21.19%) indicating very greater role played by environments and their interactions in realizing final grain yield. AMMI analysis revealed that rice hybrids viz., RNRH 39 (G6), 27P31 (G14) and RNRH 15 (G1) were recorded higher mean grain yield with positive IPCA1 scores. The hybrids, JGLH 275 (G11) and JGLH 365 (G15) were plotted near to zero IPCA1 axis indicating that these hybrids are relatively more stable across locations. GGE bi-plot genotype view depicts that the hybrids, JGLH 365 (G15) and US 314 (G8) were inside the first concentric circle and found to be more stable across environments. GGE bi-plot environment view showed that Rudrur (E4) location was the most ideal environment. However, Warangal (E6) and Jagtial (E1) locations were poor and most discriminating. Depending on dispersion of environments in different directions, six locations were partitioned into three mega zones as first zone comprised of four locations viz., Kunaram (E2), Kampasagar (E3), Rudrur (E4) and Rajendranagar (E5) whereas highly dispersed Jagtial (E1) and Warangal (E6) were identified as two separate mega environments. The bi-plot view identified that 27P31 (G14), JGL 24423 (G2) and RNRH 39 (G6) were the best performing genotypes in first zone comprising four locations.

Improvement in Ridge Coefficient Optimization Criterion for Ridge Estimation-Based Dynamic System Response Curve Method in Flood Forecasting

The ridge estimation-based dynamic system response curve (DSRC-R) method, which is an improvement of the dynamic system response curve (DSRC) method via the ridge estimation method, has illustrated its good robustness. However, the optimization criterion for the ridge coefficient in the DSRC-R method still needs further study. In view of this, a new optimization criterion called the balance and random degree criterion considering the sum of squares of flow errors (BSR) is proposed in this paper according to the properties of model-simulated residuals. In this criterion, two indexes, namely, the random degree of simulated residuals and the balance degree of simulated residuals, are introduced to describe the independence and the zero mean property of simulated residuals, respectively. Therefore, the BSR criterion is constructed by combining the sum of squares of flow errors with the two indexes. The BSR criterion, L-curve criterion and the minimum sum of squares of flow errors (MSSFE) criterion are tested on both synthetic cases and real-data cases. The results show that the BSR criterion is better than the L-curve criterion in minimizing the sum of squares of flow residuals and increasing the ridge coefficient optimization speed. Moreover, the BSR criterion has an advantage over the MSSFE criterion in making the estimated rainfall error more stable.

Sum-of-squares chordal decomposition of polynomial matrix inequalities

We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert–Artin, Reznick, Putinar, and Putinar–Vasilescu Positivstellensätze. First, we establish that a polynomial matrix P(x) with chordal sparsity is positive semidefinite for all $$x\in \mathbb {R}^n$$ x ∈ R n if and only if there exists a sum-of-squares (SOS) polynomial $$\sigma (x)$$ σ ( x ) such that $$\sigma P$$ σ P is a sum of sparse SOS matrices. Second, we show that setting $$\sigma (x)=(x_1^2 + \cdots + x_n^2)^\nu $$ σ ( x ) = ( x 1 2 + ⋯ + x n 2 ) ν for some integer $$\nu $$ ν suffices if P is homogeneous and positive definite globally. Third, we prove that if P is positive definite on a compact semialgebraic set $$\mathcal {K}=\{x:g_1(x)\ge 0,\ldots ,g_m(x)\ge 0\}$$ K = { x : g 1 ( x ) ≥ 0 , … , g m ( x ) ≥ 0 } satisfying the Archimedean condition, then $$P(x) = S_0(x) + g_1(x)S_1(x) + \cdots + g_m(x)S_m(x)$$ P ( x ) = S 0 ( x ) + g 1 ( x ) S 1 ( x ) + ⋯ + g m ( x ) S m ( x ) for matrices $$S_i(x)$$ S i ( x ) that are sums of sparse SOS matrices. Finally, if $$\mathcal {K}$$ K is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for $$(x_1^2 + \cdots + x_n^2)^\nu P(x)$$ ( x 1 2 + ⋯ + x n 2 ) ν P ( x ) with some integer $$\nu \ge 0$$ ν ≥ 0 when P and $$g_1,\ldots ,g_m$$ g 1 , … , g m are homogeneous of even degree. Using these results, we find sparse SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables, and we construct new convergent hierarchies of sparsity-exploiting SOS reformulations for convex optimization problems with large and sparse polynomial matrix inequalities. Numerical examples demonstrate that these hierarchies can have a significantly lower computational complexity than traditional ones.