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New lower bounds for matrix multiplication and $\operatorname {det}_3$
- Part of
- Austin Conner, Alicia Harper, J.M. Landsberg
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- Journal:
- Forum of Mathematics, Pi / Volume 11 / 2023
- Published online by Cambridge University Press:
- 29 May 2023, e17
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Let $M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$ denote the matrix multiplication tensor (and write $M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$ ), and let $\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$ denote the determinant polynomial considered as a tensor. For a tensor T, let $\underline {\mathbf {R}}(T)$ denote its border rank. We (i) give the first hand-checkable algebraic proof that $\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$ , (ii) prove $\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$ and $\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$ , where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was $M_{\langle 2\rangle }$ , (iii) prove $\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$ , (iv) prove $\underline {\mathbf {R}}(\operatorname {det}_3)=17$ , improving the previous lower bound of $12$ , (v) prove $\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$ for all $\mathbf {n}\geq 25$ , where previously only $\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$ was known, as well as lower bounds for $4\leq \mathbf {n}\leq 25$ , and (vi) prove $\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$ for all $\mathbf {n} \ge 18$ , where previously only $\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$ was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors.
The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, called border apolarity developed by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensor T and an integer r, in a finite number of steps, either outputs that there is no border rank r decomposition for T or produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable when T has a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory.
Spatiotemporal monitoring of hydrilla [Hydrilla verticillata (L. f.) Royle] to aid management actions
- Abhishek Kumar, Christopher Cooper, Caren M. Remillard, Shuvankar Ghosh, Austin Haney, Frank Braun, Zachary Conner, Benjamin Page, Kenneth Boyd, Susan Wilde, Deepak R. Mishra
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- Journal:
- Weed Technology / Volume 33 / Issue 3 / June 2019
- Published online by Cambridge University Press:
- 16 May 2019, pp. 518-529
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Hydrilla is an invasive aquatic plant that has rapidly spread through many inland water bodies across the globe by outcompeting native aquatic plants. The negative impacts of hydrilla invasion have become a concern for water resource management authorities, power companies, and environmental scientists. The early detection of hydrilla infestation is very important to reduce the costs associated with control and removal efforts of this invasive species. Therefore, in this study, we aimed to develop a tool for rapid, frequent, and large-scale monitoring and predicting spatial extent of hydrilla habitat. This was achieved by integrating in situ and Landsat 8 Operational Land Imager satellite data for Lake J. Strom Thurmond, the largest US Army Corps of Engineers lake east of the Mississippi River, located on the border of Georgia and South Carolina border. The predictive model for presence of hydrilla incorporated radiometric and physical measurements, including remote-sensing reflectance, Secchi disk depth (SDD), light-attenuation coefficient (Kd), maximum depth of colonization (Zc), and percentage of light available through the water column (PLW). The model-predicted ideal habitat for hydrilla featured high SDD, Zc, and PLW values, low values of Kd. Monthly analyses based on satellite images showed that hydrilla starts growing in April, reaches peak coverage around October, begins retreating in the following months, and disappears in February. Analysis of physical and meteorological factors (i.e., water temperature, surface runoff, net inflow, precipitation) revealed that these parameters are closely associated with hydrilla extent. Management agencies can use these results not only to plan removal efforts but also to evaluate and adapt their current mitigation efforts.