Distinct distances between a collinear set and an arbitrary set of points

We consider the number of distinct distances between two finite sets of points in ${\mathbb{R}}^k$, for any constant dimension $k\ge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O\left(1\right)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then

$\Omega \left(min\left\{n^{2∕3}{m}^{2∕3},\frac{n^{10∕11}{m}^{4∕11}}{{log}^{2∕11}m},n^2,m^2\right\}\right).$

Without the assumption on $P_2$, there exist sets $P_1$, $P_2$ as above, with only $O(m+n)$ distinct distances between them.     

The Elekes–Szabó Theorem in four dimensions

Let F ∈ C[x, y, s, t] be an irreducible constant-degree polynomial, and let A,B,C,D ? C be finite sets of size n. We show that F vanishes on at most O(n^8/3) points of the Cartesian product A × B × C × D, unless F has a special group-related form. A similar statement holds for A,B,C,D of unequal sizes, with a suitably modified bound on the number of zeros. This is a four-dimensional extension of our recent improved analysis of the original Elekes–Szabó theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.     

Singly Periodic Solutions of the Allen-Cahn Equation and the Toda Lattice

The Allen-Cahn equation ? Δu = u ? u ³ in ?² has family of trivial singly periodic solutions that come from the one dimensional periodic solutions of the problem ?u″ =u ? u ³. In this paper we construct a non-trivial family of singly periodic solutions to the Allen-Cahn equation. Our construction relies on the connection between this equation and the infinite Toda lattice. We show that for each one-soliton solution to the infinite Toda lattice we can find a singly periodic solution to the Allen-Cahn equation, such that its level set is close to the scaled one-soliton. The solutions we construct are analogues of the family of Riemann minimal surfaces in ?³.     

Strong negation in intuitionistic style sequent systems for residuated lattices

We study the sequent system mentioned in the author's work 18 as CyInFL with ‘intuitionistic’ sequents. We explore the connection between this system and symmetric constructive logic of Zaslavsky 40 and develop an algebraic semantics for both of them. In contrast to the previous work, we prove the strong completeness theorem for CyInFL with ‘intuitionistic’ sequents and all of its basic variants, including variants with contraction. We also show how the defined classes of structures are related to cyclic involutive FL‐algebras and Nelson FL_ew‐algebras. In particular, we prove the definitional equivalence of symmetric constructive FL_ewc‐algebras (algebraic models of symmetric constructive logic) and Nelson FL_ew‐algebras (algebras introduced by Spinks and Veroff 33 , 34 as the termwise equivalent definition of Nelson algebras). Because of the strong completeness theorem that covers all basic variants of CyInFL with ‘intuitionistic’ sequents, we rename this sequent system to symmetric constructive full Lambek calculus (). We verify the decidability of this system and its basic variants, as we did in the case of their distributive cousins 18 . As a consequence we obtain that the corresponding theories of (distributive and nondistributive) symmetric constructive FL‐algebras are decidable.     

Projection-Iterative Methods for a Class of Difference Equations

We develop a theoretical framework for projection-iterative methods to solve operator equations of the form Au + Bu = f, where A is a Toeplitz operator in a Banach space , B is considered as a perturbation (of general form) of A, and f is a given element in this space. The methods are adopted for application to general situations, in particular, to the equations in which A need not be a Fredholm operator. The idea to involve iteration procedures and the technique which we apply allow to obtain conditions on perturbations for convergence and effective error estimates in terms of some weighted spaces (without any restrictions on the norms for perturbations). Based on established evaluations we derive further information about decaying properties of the solutions. The obtained results are illustrated by considering concrete classes of equations as, for instance, equations corresponding to Jacobi type operators.      

The 1‐2‐3‐Conjecture for Hypergraphs

A weighting of the edges of a hypergraph is called vertex‐coloring if the weighted degrees of the vertices yield a proper coloring of the graph, i.e. every edge contains at least two vertices with different weighted degrees. In this article, we show that such a weighting is possible from the weight set for all hypergraphs with maximum edge size and not containing edges solely consisting of identical vertices. The number is best possible for this statement.     

α-phosphoryl sulfoxides. X. A general synthesis of optically active α-chlorovinyl sulfoxides

A general and efficient synthesis of enantiomeric α-chlorovinyl p-tolyl sulfoxides 1 using (+)-(S)_C(S)_S-α-chloro-α-dimethoxyphosphorylmethyl p-tolyl sulfoxide as a key substrate for the Horner-Wittig reaction with carbonyl compounds is described. The ratio of geometrical isomers of 1 was determined and briefly investigated. The X-ray diffraction structures of (Z)-1-chloro-1-p-tolylsulfinyl-2-phenyl-ethene and (Z)-1-chloro-1-p-tolylsulfinyl-2-(2-thienyl)-ethene are reported.     

Simultaneous determination of mitotane,its metabolite,and five steroid hormones in urine samples by capillary electrophoresis using β-CD2SDS1 complexes as hydrophobic compounds solubilizers

Mitotane is a cytotoxic drug used in the treatment of inoperable adrenocortical carcinoma, it inhibits steroidogenesis as well, and therefore monitoring the level of steroid hormones in patients treated with mitotane is a crucial point of therapy. Hence, we have developed a simple, fast, and efficient electrophoretic method combined with reverse polarity sweeping as online preconcentration technique and dispersive liquid–liquid microextraction for the simultaneous determination of mitotane, its main metabolite DDA, and five steroid hormones (progesterone, testosterone, epitestosterone, cortisol, and corticosterone) in urine samples. In addition, a new sample matrix consisting of β-CD₂SDS₁ complexes for a high hydrophobic compounds solubilization was developed. Approach based on the application of β-cyclodextrin and SDS complex of a ratio 2:1 allowed for hydrodynamic injection into the capillary of a solution containing both mitotane and other analytes. The detection limits of the analytes for the reverse polarity sweeping-dispersive liquid–liquid microextraction method were found to be in the range of 1.5–3 ng/mL, which were approximately 1000 times lower than in the conventional hydrodynamic injection (5 s, 0.5 psi) without any preconcentration procedure. All analytes were completely resolved in less than 13 min by uncoated silica capillary with an inner diameter of 75 μm (ID) × 60 cm. Electrophoretic separation was performed in reverse polarity with a voltage of –25 kV with a background electrolyte (BGE) consisting of 100 mM SDS, 25% ACN, 25 mM phosphate buffer (pH 2.5), and 7 mM β-cyclodextrin.