Related papers: Regularized Products over arithmetic Schemes
Let $P \subset \mathbb R^2$ be a point set with cardinality $N$. We give an improved bound for the number of dot products determined by $P$, proving that, \[ |\{ p \cdot q :p,q \in P \}| \gg N^{2/3+c}. \] A crucial ingredient in the proof…
Different notions on regularity of sets and of collection of sets play an important role in the analysis of the convergence of projection algorithms in nonconvex scenarios. While some projection algorithms can be applied to feasibility…
The regularized product of the Fibonacci numbers is evaluated.
Finite-part integration is a recent method of evaluating a convergent integral in terms of the finite-parts of divergent integrals deliberately induced from the convergent integral itself [E. A. Galapon, Proc. R. Soc., A 473, 20160567…
Generalizations of linear numeration systems in which the set of natural numbers is recognizable by finite automata are obtained by describing an arbitrary infinite regular language following the lexicographic ordering. For these systems of…
Observing that the logarithm of a product of two elliptic operators differs from the sum of the logarithms by a finite sum of operator brackets, we infer that regularised traces of this difference are local as finite sums of noncommutative…
We consider regularization methods based on the coupling of Tikhonov regularization and projection strategies. From the resulting constraint regularization method we obtain level set methods in a straight forward way. Moreover, we show that…
We will introduce a regularization for $p$-adic multiple zeta values and show that the generalized double shuffle relations hold. This settles a question raised by Deligne, given as a project in Arizona winter school 2002. Our approach is…
In this article we propose a general method of obtaining infinite sums of products with functions that count patterns in numbers.
We propose and analyze a regularization approach for structured prediction problems. We characterize a large class of loss functions that allows to naturally embed structured outputs in a linear space. We exploit this fact to design…
Determining the crossing numbers of Cartesian products of small graphs with arbitrarily large paths has been an ongoing topic of research since the 1970s. Doing so requires the establishment of coincident upper and lower bounds; the former…
Abstract numeration systems encode natural numbers using radix ordered words of an infinite regular language and linear recurrence sequences play a key role in their valuation. Sequence automata, which are deterministic finite automata with…
We compute the $K$-theory of crossed products of rotation algebras $\mathcal{A}_\theta$, for any real angle $\theta$, by matrices in $\mathrm{SL}(2,\mathbb{Z})$ with infinite order. Using techniques of continuous fields, we show that the…
Using exact computer arithmetic, it is possible to determine the (exact) solution of a numerical model without rounding error. For such purposes, a corresponding system of equations should be exactly defined, either directly or by…
As a generalization of [KMW], we introduce a higher Riemann zeta function for an abstract sequence. Then we explicitly determine its regularized product expression.
We give a necessary and sufficient smoothness condition for the scheme parameterizing the n-dimensional representations of a finitely generated associative algebra over an algebraically closed field of characteristic zero. In particular,…
According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of…
In this paper we show that the iterated integrals on products of one variable multiple polylogarithms from 0 to 1 are actually multiple zeta values if they are convergent. In the divergent case, we define regularized iterated integrals from…
We discuss a systematic way to dimensionally regularize divergent sums arising in field theories with an arbitrary number of physical compact dimensions or finite temperature. The method preserves the same symmetries of the action as the…
Typically, the sequence of points generated by an optimization algorithm may have multiple limit points. Under convexity assumptions, however, (sub)gradient methods are known to generate a convergent sequence of points. In this paper, we…