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We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and…
Families of operator identities appeared as a consequence of an existence of finite-dimensional representation of (super) Lie algebras of first-order differential operators and $q$-deformed (quantum) algebras of first-order…
For over a decade, the hypercomputation movement has produced computational models that in theory solve the algorithmically unsolvable, but they are not physically realizable according to currently accepted physical theories. While…
The study of the fundamental limits of information systems is a central theme in information theory. Both the traditional analytical approach and the recently proposed computational approach have significant limitations, where the former is…
A fundamental challenge in reasoning is navigating hypothetical, counterfactual worlds where logic may conflict with ingrained knowledge. We investigate this frontier for Large Language Models (LLMs) by asking: Can LLMs reason logically…
Despite great performance on Olympiad-level reasoning problems, frontier large language models can still struggle on high school math when presented with novel problems outside standard benchmarks. Going beyond final accuracy, we propose a…
Based upon elements of the modern Pseudoanalytic Function Theory, we analyse a new method for numerically approaching the solution of the Dirichlet boundary value problem, corresponding to the two-dimensional Electrical Impedance Equation.…
Hallucination, posed as a pervasive challenge of multi-modal large language models (MLLMs), has significantly impeded their real-world usage that demands precise judgment. Existing methods mitigate this issue with either training with…
This paper lies in the intersection of several fields: number theory, lattice theory, multilinear algebra, and scientific computing. We adapt existing solution algorithms for tensor eigenvalue problems to the tensor-train framework. As an…
We study the satisfiability problem of symbolic finite automata and decompose it into the satisfiability problem of the theory of the input characters and the monadic second-order theory of the indices of accepted words. We use our…
We prove level-by-level upper and lower bounds on the strength of determinacy for finite differences of sets in the hyperarithmetical hierarchy in terms of subsystems of finite-and transfinite-order arithmetic, extending the…
As reasoning LLMs increasingly trade tokens for accuracy through deliberation, search, and self-correction, a single accuracy score can no longer tell whether those tokens buy useful reasoning, recovery from hard instances, or unnecessary…
We prove a uniqueness theorem for a large class of functional equations in the plane, which resembles in form a classical result of Aczel. It is also shown that functional equations in this class are overdetermined in the sense of Paneah.…
Determining whether a program terminates is a central problem in computer science. Turing's Halting Problem established termination as undecidable, showing that no algorithm can universally determine termination for all programs and inputs.…
We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge-Amp\`ere operator on a bounded strongly pseudoconvex domain in $\C^n$. We show that the eigenfunction is…
Inequality proving, crucial across diverse scientific and mathematical fields, tests advanced reasoning skills such as discovering tight bounds and strategic theorem application. This makes it a distinct, demanding frontier for large…
The strengthening of linear relaxations and bounds of mixed integer linear programs has been an active research topic for decades. Enumeration-based methods for integer programming like linear programming-based branch-and-bound exploit…
We consider approximation or recovery of functions based on a finite number of function evaluations. This is a well-studied problem in optimal recovery, machine learning, and numerical analysis in general, but many fundamental insights were…
Humans do not just find mistakes after the fact -- we often catch them mid-stream because 'reflection' is tied to the goal and its constraints. Today's large language models produce reasoning tokens and 'reflective' text, but is it…
We consider the problem of recovering a subhypergraph based on an observed adjacency tensor corresponding to a uniform hypergraph. The uniform hypergraph is assumed to contain a subset of vertices called as subhypergraph. The edges…