Few associative triples,isotopisms and groups

Let Q be a quasigroup. For \(\alpha ,\beta \in S_Q\) let \(Q_{\alpha ,\beta }\) be the principal isotope \(x*y = \alpha (x)\beta (y)\). Put \(\mathbf a(Q)= |\{(x,y,z)\in Q^3;\) \(x(yz)) = (xy)z\}|\) and assume that \(|Q|=n\). Then \(\sum _{\alpha ,\beta }\mathbf a(Q_{\alpha ,\beta })/(n!)^2 = n^2(1+(n-1)^{-1})\), and for every \(\alpha \in S_Q\) there is \(\sum _\beta \mathbf a(Q_{\alpha ,\beta })/n! = n(n-1)^{-1}\sum _x(f_x^2-2f_x+n)\ge n^2\), where \(f_x=|\{y\in Q;\) \( y = \alpha (y)x\}|\). If G is a group and \(\alpha \) is an orthomorphism, then \(\mathbf a(G_{\alpha ,\beta })=n^2\) for every \(\beta \in S_Q\). A detailed case study of \(\mathbf a(G_{\alpha ,\beta })\) is made for the situation when \(G = \mathbb Z_{2d}\), and both \(\alpha \) and \(\beta \) are “natural” near-orthomorphisms. Asymptotically, \(\mathbf a(G_{\alpha ,\beta })>3n\) if G is an abelian group of order n. Computational results: \(\mathbf a(7) = 17\) and \(\mathbf a(8) \le 21\), where \(\mathbf a(n) = \min \{\mathbf a(Q);\) \( |Q|=n\}\). There are also determined minimum values for \(\mathbf a(G_{\alpha ,\beta })\), G a group of order \(\le 8\).     

Finite retracts of Priestley spaces and sectional coproductivity

Let Y and P be posets, let P be finite and connected, and let f : Y → P be a surjective monotone map. The map f can be naturally extended to a Priestley surjection ${\widehat{f} : \widehat{Y} \to P}$ which can turn out to be a retraction even if f is not. We characterize those maps f whose Priestley extensions ${\widehat{f}}$ are retractions. We then use this characterization to contrast, yet again, the behavior of cyclic and acyclic posets P insofar as their appearances in Priestley spaces are concerned. We say that P is sectionally coproductive if the Priestley surjection ${f : {\coprod} X_{i} \to P}$ , induced by Priestley surjections f _i : X _i → P, is a retraction only when at least one of the f _i ’s is a retraction. We then prove that P is sectionally coproductive exactly when it is acyclic.     

Code loops in both parities

We present equivalent definitions of code loops in any characteristic p≠0. The most natural definition is via combinatorial polarization, but we also show how to realize code loops by linear codes and as a class of symplectic conjugacy closed loops. For p odd, it is possible to define code loops via characteristic trilinear forms. Related concepts are discussed.     

A Singular Differential Equation Stemming from an Optimal Control Problem in Financial Economics

We consider the ordinary differential equation $$x^2 u''=axu'+bu-c \bigl(u'-1\bigr)^2, \quad x\in(0,x_0), $$ with $a\in\mathbb{R}, b\in\mathbb{R}$ , c>0 and the singular initial condition u(0)=0, which in financial economics describes optimal disposal of an asset in a market with liquidity effects. It is shown in the paper that if a+b<0 then no continuous solutions exist, whereas if a+b>0 then there are infinitely many continuous solutions with indistinguishable asymptotics near 0. Moreover, it is proved that in the latter case there is precisely one solution u corresponding to the choice x ₀=∞ which is such that 0≤u(x)≤x for all x>0, and that this solution is strictly increasing and concave.     

A difference between continuous and absolutely continuous norms in Banach function spaces

On examples we show a difference between a continuous and absolutely continuous norm in Banach function spaces.     

Quantification of measurement uncertainty in the sequential determination of <Superscript>210</Superscript>Pb and <Superscript>210</Superscript>Po by liquid scintillation counting and alpha-particle spectrometry

Methodologies for the quantification of measurement uncertainties associated with the determination of ²¹⁰Pb- and ²¹⁰Po-specific activities by liquid scintillation counting (LSC) and alpha-particle spectrometry are presented, and are demonstrated using the soil reference material IAEA-326. Major contributors to the combined uncertainty associated with the measurement result of ²¹⁰Pb were the uncertainties of net count rates in the ²¹⁰Pb energy region of the sample spectrum and in the ²¹⁰Bi energy region of the blank spectrum. The predominant sources of uncertainty in the measurement of ²¹⁰Po were the uncertainties of net count rates in the regions of interest of ²⁰⁹Po and ²¹⁰Po. The relative standard uncertainty of ²¹⁰Po exponentially increases with the time interval between the sampling date and the separation date of Po, and this effect is strongly dependent on the ²¹⁰Po/²¹⁰Pb activity ratio. When the specific activity of ²¹⁰Pb is much higher than that of ²¹⁰Po in the sample, the relative standard uncertainty of the ²¹⁰Po determination increases significantly within a short time interval between the sampling date (or reference date) and the separation date of Po in samples.