Equivalence of Systematic Linear Data Structures and Matrix Rigidity

Recently, Dvir, Golovnev, and Weinstein have shown that sufficiently strong lower bounds for linear data structures would imply new bounds for rigid matrices. However, their result utilizes an algorithm that requires an $NP$ oracle, and hence, the rigid matrices are not explicit. In this work, we derive an equivalence between rigidity and the systematic linear model of data structures. For the $n$-dimensional inner product problem with $m$ queries, we prove that lower bounds on the query time imply rigidity lower bounds for the query set itself. In particular, an explicit lower bound of $\omega\left(\frac{n}{r}\log m\right)$ for $r$ redundant storage bits would yield better rigidity parameters than the best bounds due to Alon, Panigrahy, and Yekhanin. We also prove a converse result, showing that rigid matrices directly correspond to hard query sets for the systematic linear model. As an application, we prove that the set of vectors obtained from rank one binary matrices is rigid with parameters matching the known results for explicit sets. This implies that the vector-matrix-vector problem requires query time $\Omega(n^{3/2}/r)$ for redundancy $r \geq \sqrt{n}$ in the systematic linear model, improving a result of Chakraborty, Kamma, and Larsen. Finally, we prove a cell probe lower bound for the vector-matrix-vector problem in the high error regime, improving a result of Chattopadhyay, Kouck\'{y}, Loff, and Mukhopadhyay.Comment: 23 pages, 1 tabl

Shattered Sets and the Hilbert Function

We study complexity measures on subsets of the boolean hypercube and exhibit connections between algebra (the Hilbert function) and combinatorics (VC theory). These connections yield results in both directions. Our main complexity-theoretic result demonstrates that a large and natural family of linear program feasibility problems cannot be computed by polynomial-sized constant-depth circuits. Moreover, our result applies to a stronger regime in which the hyperplanes are fixed and only the directions of the inequalities are given as input to the circuit. We derive this result by proving that a rich class of extremal functions in VC theory cannot be approximated by low-degree polynomials. We also present applications of algebra to combinatorics. We provide a new algebraic proof of the Sandwich Theorem, which is a generalization of the well-known Sauer-Perles-Shelah Lemma. Finally, we prove a structural result about downward-closed sets, related to the Chvatal conjecture in extremal combinatorics

Multi-objective design and operation of Solid Oxide Fuel Cell (SOFC) Triple Combined-cycle Power Generation systems: Integrating energy efficiency and operational safety

Energy efficiency is one of the main pathways for energy security and environmental protection. In fact, the International Energy Agency asserts that without energy efficiency, 70% of targeted emission reductions are not achievable. Despite this clarity, enhancing the energy efficiency introduce significant challenge toward process operation. The reason is that the methods applied for energy-saving pose the process operation at the intersection of safety constraints. The present research aims at uncovering the trade-off between safe operation and energy efficiency; an optimization framework is developed that ensures process safety and simultaneously optimizes energy-efficiency, quantified in economic terms. The developed optimization framework is demonstrated for a solid oxide fuel cell (SOFC) power generation system. The significance of this industrial application is that SOFC power plants apply a highly degree of process integration resulting in very narrow operating windows. However, they are subject to significant uncertainties in power demand. The results demonstrate a strong trade-off between the competing objectives. It was observed that highly energy-efficient designs feature a very narrow operating window and limited flexibility. For instance, expanding the safe operating window by 100% will incur almost 47% more annualized costs. Establishing such a trade-off is essential for realizing energy-saving

Multi-objective design and operation of solid oxide fuel cell (SOFC) Triple Combined-cycle Power Generation Systems: integrating energy efficiency and operational safety

Energy efficiency is one of the main pathways for energy security and environmental protection. In fact, the International Energy Agency asserts that without energy efficiency, 70% of targeted emission reductions are not achievable. Despite this clarity, enhancing the energy efficiency introduce significant challenge toward process operation. The reason is that the methods applied for energy-saving pose the process operation at the intersection of safety constraints. The present research aims at uncovering the trade-off between safe operation and energy efficiency; an optimization framework is developed that ensures process safety and simultaneously optimizes energy-efficiency, quantified in economic terms. The developed optimization framework is demonstrated for a solid oxide fuel cell (SOFC) power generation system. The significance of this industrial application is that SOFC power plants apply a highly degree of process integration resulting in very narrow operating windows. However, they are subject to significant uncertainties in power demand. The results demonstrate a strong trade-off between the competing objectives. It was observed that highly energy-efficient designs feature a very narrow operating window and limited flexibility. For instance, expanding the safe operating window by 100% will incur almost 47% more annualized costs. Establishing such a trade-off is essential for realizing energy-saving

Vector-Matrix-Vector Queries for Solving Linear Algebra, Statistics, and Graph Problems

We consider the general problem of learning about a matrix through vector-matrix-vector queries. These queries provide the value of u^{T}Mv over a fixed field ? for a specified pair of vectors u,v ? ??. To motivate these queries, we observe that they generalize many previously studied models, such as independent set queries, cut queries, and standard graph queries. They also specialize the recently studied matrix-vector query model. Our work is exploratory and broad, and we provide new upper and lower bounds for a wide variety of problems, spanning linear algebra, statistics, and graphs. Many of our results are nearly tight, and we use diverse techniques from linear algebra, randomized algorithms, and communication complexity