Related papers: Pseudojump inversion in special r. b. $\Pi^0_1$ cl…
The Posner-Robinson Theorem states that for any reals $Z$ and $A$ such that $Z \oplus 0' \leq_\mathrm{T} A$ and $0 <_\mathrm{T} Z$, there exists $B$ such that $A \equiv_\mathrm{T} B' \equiv_\mathrm{T} B \oplus Z \equiv_\mathrm{T} B \oplus…
This paper is an investigation of the relationship between G\"odel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation $\{(A,B) \in \mathbb{R}^2 :…
Tennenbaum's theorem states that PA does not admit any nonstandard computable model. In 2022, Pakhomov proved that this theorem is fragile in regards to how PA is expressed, by constructing a theory that is definitionally equivalent to PA…
We establish an It\^o-type formula for finite $p$-variation paths with jumps for arbitrary $p\geq 1$. The formula is stated in a fully pathwise form and separates the reduced rough integral from explicit left- and right-jump correction…
The Robinson Splitting Theorem states that a c.e. degree $\mathbf{b}$ splits over any low c.e. degree $\mathbf{c}<\mathbf{b}$. We prove that a weaker version of this theorem holds in models of $\mathrm{P}^-+\mathrm{I}\Sigma_1$, with lowness…
There are noncomputable c.e.\ sets, computable from every SJT-hard c.e.\ set. This yields a natural pseudo-jump operator, increasing on all sets, which cannot be inverted back to a minimal pair or even avoiding an upper cone.
We say that a structure $\mathcal{A}$ admits \emph{strong jump inversion} provided that for every oracle $X$, if $X'$ computes $D(\mathcal{C})'$ for some $\mathcal{C}\cong\mathcal{A}$, then $X$ computes $D(\mathcal{B})$ for some…
We present a generalization of the multiplier ideal version of inversion of adjunction, often known as the restriction theorem, to centers of arbitrary codimension. We approach inversion of adjunction from the subadjunction point of view.…
By investigating in detail discontinuities of the first kind of real-valued functions and the analysis of unordered sums, where the summands are given by values of a positive real-valued function, we develop a measure-theoretical framework…
In this paper we will study an important but rather technical result which is called The Reduction Property. The result tells us how much arithmetical conservation there is between two arithmetical theories. Both theories essentially speak…
We investigate N\'eron models of Jacobians of singular curves over strictly Henselian discretely valued fields, and their behaviour under tame base change. For a semiabelian variety, this behaviour is governed by a finite sequence of (a…
Erd\H{o}s conjectured in 1945 that for any unit vectors $v_1, \dotsc, v_n$ in $\mathbb{R}^2$ and signs $\varepsilon_1, \dotsc, \varepsilon_n$ taken independently and uniformly in $\{-1,1\}$, the random Rademacher sum $\sigma = \varepsilon_1…
It is a classical fact that every $n$-element set of positive reals has at least $\binom{n+1}{2}+1$ distinct subset sums, with equality exactly for homogeneous arithmetic progressions (when $n\geq 4$). We establish stability versions of…
In this note, we give a necessary and sufficient condition under which the comparison theorem holds for multidimensional stochastic differential equations (SDEs) with jumps and for matrix-valued SDEs with jumps.
We say that a real X is n-generic relative to a perfect tree T if X is a path through T and for all Sigma^0_n (T) sets S, there exists a number k such that either X|k is in S or for all tau in T extending X|k we have tau is not in S. A real…
Jump systems are sets of integer vectors satisfying a simple axiom, generalizing matroids, also delta-matroids, and well-kown combinatorial examples such as degree sequences of subgraphs of a graph. It is useful to know if a set of vectors…
Consider a vertex-reinforced jump process defined on a regular tree, where each vertex has exactly $b$ children, with $b \ge 3$. We prove the strong law of large numbers and the central limit theorem for the distance of the process from the…
The big jump principle is a well established mathematical result for sums of independent and identically distributed random variables extracted from a fat tailed distribution. It states that the tail of the distribution of the sum is the…
In this paper we establish a comparison theorem for stochastic differential delay equations with jumps. An example is constructed to demonstrate that the comparison theorem need not hold whenever the diffusion term contains a delay function…
We show that degrees containing a complete extensions of arithmetic have the random join property: they are the supremum of any random real they compute, with another random real. The same is true for the truth-table and weak truth-table…