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In this paper, we propose a generalization of Continuous Logic ([BBHU08]) where the distances take values in suitable co-quantales (in the way as it was proposed in [Fla97]). By assuming suitable conditions (e.g., being co-divisible,…
Testing the independence between random vectors is a fundamental problem in statistics. Distance correlation, a recently popular dependence measure, is universally consistent for testing independence against all distributions with finite…
Relative index theorems, which deal with what happens with the index of elliptic operators when cutting and pasting, are abundant in the literature. It is desirable to obtain similar theorems for other stable homotopy invariants, not the…
A quantitative form of the Nullity Theorem is presented, which establishes a linear relation between the singular values of the two submatrices involved in the theorem up to the first order. The theorem is then extended to function spaces…
Lawvere showed that generalised metric spaces are categories enriched over $[0, \infty]$, the quantale of the positive extended reals. The statement of enrichment is a quantitative analogue of being a preorder. Towards seeking a logic for…
We study the zero sets of the independence polynomial on recursive sequences of graphs. We prove that for a maximally independent starting graph and a stable and expanding recursion algorithm, the zeros of the independence polynomial are…
The linear complementarity problem is a continuous optimization problem that generalizes convex quadratic programming, Nash equilibria of bimatrix games and several such problems. This paper presents a continuous optimization formulation…
We define tests of boolean functions which distinguish between linear (or quadratic) polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between…
We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that…
This thesis develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and…
We revisit the results on admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements…
We obtain direct and inverse approximation theorems of functions of several variables by Taylor-Abel-Poisson means in the integral metrics. We also show that norms of multipliers in the spaces $L_{p,Y}(\mathbb T^d)$ are equivalent for all…
Probabilistic context-free grammars (PCFGs) are used to define distributions over strings, and are powerful modelling tools in a number of areas, including natural language processing, software engineering, model checking, bio-informatics,…
We study approximate equivalence relations up to commensurability, in the presence of a definable measure. As a basic framework, we give a presentation of probability logic based on continuous logic. Hoover's normal form is valid here; if…
Recent improvement on Tarski's procedure for quantifier elimination in the first order theory of real numbers makes it feasible to solve small instances of the following problems completely automatically: 1. listing all equality and…
For probability measures on a complete separable metric space, we present sufficient conditions for the existence of a solution to the Kantorovich transportation problem. We also obtain sufficient conditions (which sometimes also become…
Cut-elimination theorems constitute one of the most important classes of theorems of proof theory. Since Gentzen's proof of the cut-elimination theorem for the system $\mathbf{LK}$, several other proofs have been proposed. Even though the…
The notion of relative universality with respect to a {\sigma}-field was introduced to establish the unbiasedness and Fisher consistency of an estimator in nonlinear sufficient dimension reduction. However, there is a gap in the proof of…
We discuss the (first- and second-order) optimality conditions for nonlinear programming under the relaxed constant rank constraint qualification. This condition generalizes the so-called linear independence constraint qualification.…
For positive $q\neq1$, the $q$-exchangeability of an infinite random word is introduced as quasi-invariance under permutations of letters, with a special cocycle which accounts for inversions in the word. This framework allows us to extend…