Related papers: Half-Iterates and Delta Conjectures
We study the system of semilinear elliptic equations $$-\Delta u_i+ u_i = \sum_{j=1}^\ell \beta_{ij}|u_j|^p|u_i|^{p-2}u_i, \qquad u_i\in H^1(\mathbb{R}^N),\qquad i=1,\ldots,\ell,$$ where $N\geq 4$, $1<p<\frac{N}{N-2}$, and the matrix…
We prove a linear in $\deg\omega$ upper bound on the number of real zeros of the Abelian integral $I(t)=\int_{\delta(t)}\omega$, where $\delta(t)\subset\R^2$ is the real oval $x^2y(1-x-y)=t$ and $\omega$ is a one-form with polynomial…
There has been significant work recently on integer programs (IPs) $\min\{c^\top x \colon Ax\leq b,\,x\in \mathbb{Z}^n\}$ with a constraint marix $A$ with bounded subdeterminants. This is motivated by a well-known conjecture claiming that,…
Let the sets of functions $Z$ and $\Omega$ be given on the time interval $T$, let there also be a multifunction (m/f) $\alpha$ acting from $\Omega$ to $Z$ and a finite set of moments $\Delta$ from $T$. The work deals with two questions: the…
The demonstration and use of nonlocality, as defined by Bell's theorem, rely strongly on dealing with non-detection events due to losses and detector inefficiencies. Otherwise, the so-called detection loophole could be exploited. The only…
We evaluate the determinant $\det_{1\leq i,j\leq n}(\binom{x+y+j}{x-i+2j}-\binom{x+y+j}{x+i+2j})$, which gives the number of lozenge tilings of a hexagon with cut off corners. A particularly interesting feature of this evaluation is that it…
This paper presents a comprehensive analysis of the well-known extragradient (EG) method for solving both equations and inclusions. First, we unify and generalize EG for [non]linear equations to a wider class of algorithms, encompassing…
We investigate solutions to the functional equation $f(f(x)) = e^x$, which can be interpreted as the problem of finding a half iterate of the exponential map. While no elementary solution exists, we construct and analyze non-elementary…
In this study, a new $\Delta$-evaluation method is introduced for solving a column permutation problem defined on a sparse binary matrix with the consecutive ones property. This problem models various $\mathcal{NP}$-hard problems in graph…
Abel's functional equation for $2^{x/2}$ and half-iterates of $\lambda x (1-x)$ & $\sqrt{1+x}$ are featured in this collection of exercises ($0 < \lambda \neq 1 < 2$).
We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +C(t)x. Specifically, we find bounds on the number of its rational solutions when A(t), B(t) and C(t) are polynomials with real or complex coefficients; and on the number of…
We study alternating parity good-for-games (GFG) automata, i.e., alternating parity automata where both conjunctive and disjunctive choices can be resolved in an online manner, without knowledge of the suffix of the input word still to be…
Recently Ahmadi et al. (2021) and Tagliaferro (2022) proposed some iterative methods for the numerical solution of linear systems which, under the classical hypothesis of strict diagonal dominance, typically converge faster than the Jacobi…
The formula of expanding the Abel variety theta function restricted to Abel subvariety into theta functions of this subvariety is obtained. With the help of this formula the solution of differential equations with Jacobi theta functions,…
In this paper we investigate the parallelization of two modular algorithms. In fact, we consider the modular computation of Gr\"obner bases (resp. standard bases) and the modular computation of the associated primes of a zero-dimensional…
A sharp explicit estimate is proved for the difference $e^\beta-\alpha$ when $\alpha$ and $\beta$ are nonzero algebraic numbers.
To check the satisfiability of (non-linear) real arithmetic formulas, modern satisfiability modulo theories (SMT) solving algorithms like NLSAT depend heavily on single cell construction, the task of generalizing a sample point to a…
In this essay we'll prove G\"odel's incompleteness theorems twice. First, we'll prove them the good old-fashioned way. Then we'll repeat the feat in the setting of computation. In the process we'll discover that G\"odel's work, rightly…
Entity Resolution constitutes a core data integration task that relies on Blocking in order to tame its quadratic time complexity. Schema-agnostic blocking achieves very high recall, requires no domain knowledge and applies to data of any…
A new computational method that uses polynomial equations and dynamical systems to evaluate logical propositions is introduced and applied to Goedel's incompleteness theorems. The truth value of a logical formula subject to a set of axioms…