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A new version of classical S-procedure in system theory is proposed based on duality in the space of positive definite matrices and introduction of matrix Lagrange multipliers. A new proof and extension of the recent results of T.Iwasaki,…
Formal theorem proving, a field at the intersection of mathematics and computer science, has seen renewed interest with advancements in large language models (LLMs). This paper introduces SubgoalXL, a novel approach that synergizes…
This work is concerned with the proof-complexity of certifying that optimization problems do \emph{not} have good solutions. Specifically we consider bounded-degree "Sum of Squares" (SOS) proofs, a powerful algebraic proof system introduced…
In practice, non-specialized interior point algorithms often cannot utilize the massively parallel compute resources offered by modern many- and multi-core compute platforms. However, efficient distributed solution techniques are required,…
In this paper ISS small-gain theorems for discrete-time systems are stated, which do not require input-to-state stability (ISS) of each subsystem. This approach weakens conservatism in ISS small-gain theory, and for the class of…
In this paper, we present a numerical method for rigorously finding the monodromy of linear differential equations. Beginning at a base point where certain particular solutions are explicitly given by series expansions, we first compute the…
We propose a parametric integral probability metric (IPM) to measure the discrepancy between two probability measures. The proposed IPM leverages a specific parametric family of discriminators, such as single-node neural networks with ReLU…
While deep neural networks are sensitive to adversarial noise, sparse coding using the Basis Pursuit (BP) method is robust against such attacks, including its multi-layer extensions. We prove that the stability theorem of BP holds upon the…
Multilevel methods are among the most efficient numerical methods for solving large-scale linear systems that arise from discretized partial differential equations. The fundamental module of such methods is a two-level procedure, which…
We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ as formal…
We study multivariate trigonometric polynomials, satisfying a set of constraints close to the known Strung-Fix conditions. Based on the polyphase representation of these polynomials relative to a general dilation matrix, we develop a simple…
Algorithms for solving the linear classification problem have a long history, dating back at least to 1936 with linear discriminant analysis. For linearly separable data, many algorithms can obtain the exact solution to the corresponding…
Solving high dimensional partial differential equations (PDEs) has historically posed a considerable challenge when utilizing conventional numerical methods, such as those involving domain meshes. Recent advancements in the field have seen…
In this paper, the concept of matrix splitting is introduced to solve a large sparse ill-posed linear system via Tikhonov's regularization. In the regularization process, we convert the ill-posed system to a well-posed system. The…
We propose an exactly solvable self-consistent kinetic model of polymerization-induced phase separation (PIPS) via spinodal decomposition. Using modified Cahn-Hilliard and Glotzer-Coniglio theories for early and late stages of spinodal…
This paper focuses on proposing a deep learning initialized iterative method (Int-Deep) for low-dimensional nonlinear partial differential equations (PDEs). The corresponding framework consists of two phases. In the first phase, an…
We present a self-contained separation framework for P vs NP developed entirely within ZFC. The approach consists of: (i) a deterministic, radius-1 compilation from uniform polynomial-time Turing computation to local sum-of-squares (SoS)…
A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic black-box identity testing algorithm for…
A block alternating splitting iteration (BASI) method is presented for solving the system arising from the finite element discretization of the distributed optimal control problem with time-periodic parabolic equations. We prove that the…
In this work, we give a novel general approach for distribution testing. We describe two techniques: our first technique gives sample-optimal testers, while our second technique gives matching sample lower bounds. As a consequence, we…