On sayings and teachings of 11th century Fatimid Caliph.

Kerekh 1. shaʼar 1 pereḳ 1-6 -- Kerekh 2. shaʻar 1 pereḳ 7-22.

Maran ha-g. R. A. Y. L. Shṭinman z. ts. ṿe-ḳ.l. -- Rabotenu meʼore ha-dor -- Ḳiyum ha-ʻolam ʻa. y. Torah be-mishnat rabotenu -- Maran Rabi Gedalyah Aizman z. ts. ṿe-ḳ.l.

The energy-spectrum of two point-like particles interacting in a 3-D isotropic Harmonic Oscillator (H.O.) trap is related to the free scattering phase-shifts \(\delta \) of the particles by a formula first published by Busch et al. It is here used to find an expression for the shift of the energy levels, caused by the interaction, rather than the perturbed spectrum itself. In the limit of high energy (large quantum number \(n\) of the H.O.) this shift (in H.O. units) is shown (...) to be given by \(\Delta =-2\frac{\delta }{\pi }\) , also exact in the limit of infinite scattering length ( \(\delta =\pm \frac{\pi }{2}\) ) in which case \(\Delta =\mp 1\) . Numerical investigation shows that this expression otherwise differs from the exact result of Busch et al., by less than \(\frac{1}{2}\,\%\) except for \(n=0\) when it can be as large as \(\approx \) 2.5 %. This result for the energy-shift is well known from another exactly solvable model, namely that of two particles interacting in a spherical infinite square-well trap (or box) of radius \(R\) in the limit \(R\rightarrow \infty \) , and/or in the limit of large energy. It is in solid state physics referred to as Fumi’s theorem. It can be (and has been) used in (infinite) nuclear matter calculations to calculate the two-body effective interaction in situations where in-medium effects can be neglected. It is in this context referred to as the phase-shift approximation a term also used throughout this report. (shrink)

Heisenberg's position-measurement-momentum-disturbance relation is derivable from the uncertainty relation σ(q)σ(p) ≥ h/2 only for the case when the particle is initially in a momentum eigenstate. Here I derive a new measurement-disturbance relation which applies when the particle is prepared in a twin-slit superposition and the measurement can determine at which slit the particle is present. The relation is d × Δp ≥ 2h/π, where d is the slit separation and Δp = DM(Pf, Pi) is the Monge distance between the initial (...) Pi(p) and final Pf(p) momentum distributions. (shrink)

Background Informed consent is a requirement for all research. It is not, however, clear how much information is sufficient to make an informed decision about participation in research. Information on an online questionnaire about childhood development was provided through an unfolding electronic participant sheet in three levels of information. Methods 552 participants, who completed the web-based survey, accessed and spent time reading the participant information sheet (PIS) between July 2008 and November 2009. The information behaviour of the participants was investigated. (...) The first level contained less information than might be found on a standard PIS, the second level corresponded to a standard PIS, and the third contained more information than on a standard PIS. The actual time spent on reading the information provided in three incremental levels and the participants' evaluation of the information were calculated. Results 77% of the participants chose to access the first level of information, whereas 12% accessed the first two levels, 6% accessed all three levels of information and 23% participated without accessing information. The most accessed levels of information were those that corresponded to the average reading times. Conclusion The brief information provided in the first level was sufficient for participants to make informed decisions, while a sizeable minority of the participants chose not to access any information at all. This study adds to the debate about how much information is required to make a decision about participation in research and the results may help inform the future development of information sheets by providing data on participants' actual needs when deciding about questionnaire surveys. (shrink)

In Hermathena, vol. xiv., pp. 338–359, Professor Charles Exon attempts to prove that the nom. and dat.-abl. pi. of deus were disyllabic in Plautus. The argument upon which he lays most stress is briefly this: Plautus uses iambic words shortened by the law of breves breviantes in the thesis of iambic and trochaic verse about twice as often as he does in the arsis, whereas the long monosyllable cor occurs in dialogue with equal frequency in both parts of the foot. (...) In this respect the nom. pl. of deus agrees with the shortened iambi, and should therefore, he thinks, be considered one of them. (shrink)

Recall that B is PA relative to A if B computes a member of every nonempty $\Pi ^0_1(A)$ class. This two-place relation is invariant under Turing equivalence and so can be thought of as a binary relation on Turing degrees. Miller and Soskova [23] introduced the notion of a $\Pi ^0_1$ class relative to an enumeration oracle A, which they called a $\Pi ^0_1{\left \langle {A}\right \rangle }$ class. We study the induced extension of the relation B is PA relative (...) to A to enumeration oracles and hence enumeration degrees. We isolate several classes of enumeration degrees based on their behavior with respect to this relation: the PA bounded degrees, the degrees that have a universal class, the low for PA degrees, and the ${\left \langle {\text {self}\kern1pt}\right \rangle }$ -PA degrees. We study the relationship between these classes and other known classes of enumeration degrees. We also investigate a group of classes of enumeration degrees that were introduced by Kalimullin and Puzarenko [14] based on properties that are commonly studied in descriptive set theory. As part of this investigation, we give characterizations of three of their classes in terms of a special sub-collection of relativized $\Pi ^0_1$ classes—the separating classes. These three can then be seen to be direct analogues of three of our classes. We completely determine the relative position of all classes in question. (shrink)

In the context of computational social choice, we study voting methods that assign a set of winners to each profile of voter preferences. A voting method satisfies the property of positive involvement (PI) if for any election in which a candidate x would be among the winners, adding another voter to the election who ranks x first does not cause x to lose. Surprisingly, a number of standard voting methods violate this natural property. In this paper, we investigate different ways (...) of measuring the extent to which a voting method violates PI, using computer simulations. We consider the probability (under different probability models for preferences) of PI violations in randomly drawn profiles vs. profile-coalition pairs (involving coalitions of different sizes). We argue that in order to choose between a voting method that satisfies PI and one that does not, we should consider the probability of PI violation conditional on the voting methods choosing different winners. We should also relativize the probability of PI violation to what we call voter potency, the probability that a voter causes a candidate to lose. Although absolute frequencies of PI violations may be low, after this conditioning and relativization, we see that under certain voting methods that violate PI, much of a voter's potency is turned against them - in particular, against their desire to see their favorite candidate elected. (shrink)

Suppose that M⊧PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}\models \mathsf{PA}$$\end{document} and X⊆P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak X} \subseteq {\mathcal P}$$\end{document}. If M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}$$\end{document} has a finitely generated elementary end extension N≻endM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal N}\succ _\mathsf{end} {\mathcal M}$$\end{document} such that {X∩M:X∈Def}=X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{X \cap M : X \in {{\mathrm{Def}}}\} = (...) {\mathfrak X}$$\end{document}, then there is such an N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal N}$$\end{document} that is, in addition, a minimal extension of M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}$$\end{document} iff every subset of M that is Π10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _1^0$$\end{document}-definable in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} is the countable union of Σ10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _1^0$$\end{document}-definable sets. (shrink)

The present study aims to examine the relationship between the cortical midline structures (CMS), which have been regarded to be associated with selfhood, and moral decision making processes at the neural level. Traditional moral psychological studies have suggested the role of moral self as the moderator of moral cognition, so activity of moral self would present at the neural level. The present study examined the interaction between the CMS and other moral-related regions by conducting psycho-physiological interaction analysis of functional images (...) acquired while 16 subjects were solving moral dilemmas. Furthermore, we performed Granger causality analysis to demonstrate the direction of influences between activities in the regions in moral decision-making. We first demonstrate there are significant positive interactions between two central CMS seed regions—i.e., the medial prefrontal cortex (MPFC) and posterior cingulate cortex (PCC)—and brain regions associated with moral functioning including the cerebellum, brainstem, midbrain, dorsolateral prefrontal cortex, orbitofrontal cortex and anterior insula (AI); on the other hand, the posterior insula (PI) showed significant negative interaction with the seed regions. Second, several significant Granger causality was found from CMS to insula regions particularly under the moral-personal condition. Furthermore, significant dominant influence from the AI to PI was reported. Moral psychological implications of these findings are discussed. The present study demonstrated the significant interaction and influence between the CMS and morality-related regions while subject were solving moral dilemmas. Given that, activity in the CMS is significantly involved in human moral functioning. (shrink)

Socio-philosophical work in response to the philosophy of Pitirim Aleksandrovich Sorokin.

We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form $$\phi ^\mathcal {M}\le r$$, and the open diagram, which encapsulates strict inequalities of the form $$\phi ^\mathcal {M}< r$$. We show (...) that the closed and open $$\Sigma _N$$ diagrams are $$\Pi ^0_{N+1}$$ and $$\Sigma ^0_N$$ respectively, and that the closed and open $$\Pi _N$$ diagrams are $$\Pi ^0_N$$ and $$\Sigma ^0_{N + 1}$$ respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal. (shrink)

Mavo be-mishnat ha-Ḥasidut, emunat H., biṭaḥon ṿe-hishtadlut, emunat tsadiḳim, hitḳashrut la-tsadiḳim, ahavat ḥaverim, tiḳun ha-midot, hakhanah le-mitsṿah, Torah, tefilah, teshuvah, be-khol derakhekha daʻehu, Shabat, ʻanaṿah, śimḥah.

We show that a theory of autonomous iterated Ramseyness based on second order arithmetic (SOA) is proof-theoretically equivalent to ${\Pi^1_2}$ -comprehension. The property of Ramsey is defined as follows. Let X be a set of real numbers, i.e. a set of infinite sets of natural numbers. We call a set H of natural numbers homogeneous for X if either all infinite subsets of H are in X or all infinite subsets of H are not in X. X has the property (...) of Ramsey if there exists a set which is homogeneous for X. The property of Ramsey is considered in reverse mathematics to compare the strength of subsystems of SOA. To characterize the system of ${\Pi^1_2}$ -comprehension in terms of Ramseyness we introduce a system of autonomous iterated Ramseyness, called R-calculus. We augment the language of SOA with additional set terms (called R-terms) ${R\vec{x}X\phi(\vec{x},X)}$ for each first order formula ${\phi(\vec{x},X)}$ (where φ may contain further R-terms). The R-calculus is a system which comprises comprehension for all first order formulas (which may contain R-terms or other set parameters) and defining axioms for the R-terms which claim that for each ${\vec{x}}$ , we can remove finitely many elements from the set ${R\vec{x}X\phi(\vec{x},X)}$ such that the remaining set is homogeneous for ${\{{X}{\phi(\vec{x},X)\}}}$ . We show that the R-calculus proves the same ${\Pi^1_1}$ -sentences as the system of ${\Pi^1_2}$ -comprehension. (shrink)

[1] Torah ṿe-yirʼat Shamayim -- [2] Midot ṿe-hanhagot ṭovot.