Related papers: ${\Sigma}^0_3$ determinacy and ${\mathbf{\Pi}}^1_2…

We describe a model $\mathcal{M}_3^{comb}$ for the boundary of the connectedness locus $\mathcal{M}^{sy}_3$ of the parameter space of cubic symmetric polynomials $p_c(z)=z^3-3c^2z$. We show that there exists a monotone continuous function…

Hemaspaandra et al. proved that, for $m > 0$ and $0 < i < k - 1$: if $\Sigma_i^p \BoldfaceDelta DIFF_m(\Sigma_k^p)$ is closed under complementation, then $DIFF_m(\Sigma_k^p) = coDIFF_m(\Sigma_k^p)$. This sharply asymmetric result fails to…

The idea of monotonicity (or positive-definiteness in the linear case) is shown to be the central theme of the solution theories associated with problems of mathematical physics. A "grand unified" setting is surveyed covering a…

We prove the (generalized) principal pivot transform is matrix monotone, in the sense of the L\"owner ordering, under minimal hypotheses. This improves on the recent results of J. E. Pascoe and R. Tully-Doyle, Monotonicity of the principal…

This paper presents a cut-elimination proof for the logic $LG^\omega$, which is an extension of a proof system for encoding generic judgments, the logic $\FOLDNb$ of Miller and Tiu, with an induction principle. The logic $LG^\omega$, just…

Assume $\boldsymbol{\Delta}^1_{2}$-determinacy. Let $L_{\kappa_3}[T_2]$ be the admissible closure of the Martin-Solovay tree and let $M_{1,\infty}$ be the direct limit of $M_1$ via countable trees. We show that $L_{\kappa_3}[T_2] \cap…

We study the Sierpinski object $\Sigma$ in the realizability topos based on Scott's graph model of the $\lambda$-calculus. Our starting observation is that the object of realizers in this topos is the exponential $\Sigma ^N$, where $N$ is…

We prove the crepant resolution conjecture for Donaldson-Thomas invariants of hard Lefschetz CY3 orbifolds, formulated by Bryan-Cadman-Young, interpreting the statement as an equality of rational functions. In order to do so, we show that…

In this paper, we study the employment of $\Sigma_1$-sentences with certificates, i.e., $\Sigma_1$-sentences where a number of principles is added to ensure that the witness is sufficiently number-like. We develop certificates in some…

We show that results of Akama, Berardi, Hayashi and Kohlenbach, on the relative independence of certain arithmetical principles over intuitionistic arithmetic HA, hold also over Kleene and Vesley's system FIM of intuitionistic analysis,…

In this paper, we prove a congruence which confirms a conjecture of Adamchuk. For any prime $p\equiv1\pmod3$ and $a\in\mathbb{Z}^{+}$, we have \begin{align*} \sum_{k=1}^{\frac{2}3(p^a-1)}\binom{2k}k\equiv0\pmod{p^2}. \end{align*}

Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…

We prove a motivic enhancement of the classical Picard--Lefschetz formula. Our proof is completely motivic, and yields a description of the motivic nearby cycles at a quasi-homogeneous singularity, as well as its monodromy, in terms of an…

We study the dynamics of $SL_3(\mathbb{R})$ and its subgroups on the homogeneous space $X$ consisting of homothety classes of rank-2 discrete subgroups of $\mathbb{R}^3$. We focus on the case where the acting group is Zariski dense in…

We study a relation between three different formulations of theorems on separable determination - one using the concept of rich families, second via the concept of suitable models and third, a new one, suggested in this paper, using the…

In this paper, we study the Cauchy problem for the Chern-Simons gauged $O(3)$ sigma model under the Lorenz gauge condition. We prove the local well-posedness of solutions if the initial matter field and gauge field satisfy $(\bm{\phi}_0,…

We prove effective versions of Oppenheim's conjecture for generic inhomogeneous forms in the S-arithmetic setting. We prove an effective result for fixed rational shifts and generic forms and we also prove a result where both the quadratic…

We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over $\mathbb{C}$ attached to character polynomials. Our approach interprets the stabilization as a probabilistic…

Let $E$ be a uniformly smooth and uniformly convex real Banach space and $E^*$ be its dual space. Suppose $A : E\rightarrow E^*$ is bounded, strongly monotone and satisfies the range condition such that $A^{-1}(0)\neq \emptyset$. Inspired…

We prove that the monodromy of an irreducible cohomologically complex rigid local system with finite determinant and quasi-unipotent local monodromies at infinity on a smooth quasiprojective complex variety $X$ is integral. This answers…