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This monograph elucidates and extends many theorems and conjectures in analytic number theory and algebraic asymptotic analysis via the natural notion of "degree" and a more general notion that we call "logexponential degree." Specifically,…
We develop analytical methods for nonlinear Dirac equations. Examples of such equations include Dirac-harmonic maps with curvature term and the equations describing the generalized Weierstrass representation of surfaces in three-manifolds.…
Special classes of analytic functions, denoting by K and J arc considered in this paper.
For a nonempty compact subset $\sigma$ in the plane, the space $AC(\sigma)$ is the closure of the space of complex polynomials in two real variables under a particular variation norm. In the classical setting, $AC[0,1]$ contains several…
We define a new type of valuation of a ring that combines the notion of Krull valuation with that of multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations. This…
We propose the Lie-algebraic interpretation of poly-analytic functions in $L_2(\C,d\mu)$, with the Gaussian measure $d\mu$, based on a flag structure formed by the representation spaces of the $\mathfrak{sl}(2)$-algebra realized by…
The polyadic mu-calculus is a modal fixpoint logic whose formulas define relations of nodes rather than just sets in labelled transition systems. It can express exactly the polynomial-time computable and bisimulation-invariant queries on…
In this paper, we show the arc length of the optimal ROC curve is an $f$-divergence. By leveraging this result, we express the arc length using a variational objective and estimate it accurately using positive and negative samples. We show…
We introduce the Arithmetic Site: an algebraic geometric space deeply related to the non-commutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient of the adele class space of the field of…
Let $X$ be a separable Banach space with a separating polynomial. We show that there exists $C\geq 1$ (depending only on $X$) such that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and every $\epsilon>0$, there exists a…
We formulate a resolution of singularities algorithm for analyzing the zero sets of real-analytic functions in dimensions $\geq 3$. Rather than using the celebrated result of Hironaka, the algorithm is modeled on a more explicit and…
In this paper we argue that the well-known maximal extensions of the Kerr and Kerr-Newman spacetimes characterized by a specific gluing (on disks) of two asymptotically flat regions with ADM masses of opposite signs are physically…
This note surveys basic topological properties of nonarchimedean analytic spaces, in the sense of Berkovich, including the recent tameness results of Hrushovski and Loeser. We also discuss interactions between the topology of nonarchimedean…
A space is `n-arc connected' (n-ac) if any family of no more than n-points are contained in an arc. For graphs the following are equivalent: (i) 7-ac, (ii) n-ac for all n, (iii) continuous injective image of a closed sub-interval of the…
Working in the p-adic analog of the complex numbers, we'll define a line integral on a small arc of a circle. This allows new versions of the Residue Theorem, the Cauchy-Goursat Theorem on discs with and without holes, Cauchy's Integral…
In this paper, we give a description of the possible poles of the local zeta function attached to a complex or real analytic mapping in terms of a log-principalization of an ideal associated to the mapping. When the mapping is a…
We show that, given a family of discs centered at a chord-arc curve, the analytic capacity of a union of arbitrary subsets of these discs (one subset in each disc) is comparable with the sum of their analytic capacities. We show a sort of…
We prove that the zero locus of an admissible normal function over an algebraic parameter space S is algebraic in the case where S is a curve.
A standard theorem in nonsmooth analysis states that a piecewise affine function $F:\mathbb R^n\rightarrow\mathbb R^n$ is surjective if it is coherently oriented in that the linear parts of its selection functions all have the same nonzero…
We present several results on the compactness of the space of morphisms between analytic spaces in the sense of Berkovich. We show that under certain conditions on the source, every sequence of analytic maps having an affinoid target has a…