A1 Either a property or its negation is positive, but not both: ∀φ[P (¬φ)↔ ¬P (φ)] A2 A property necessarily implied by a positive property is positive: ∀φ∀ψ[(P (φ) ∧ ∀x[φ(x)→ ψ(x)])→ P (ψ)] T1 Positive properties are possibly exemplified: ∀φ[P (φ)→ ♦∃xφ(x)] D1 A God-like being possesses all positive properties: G(x)↔ ∀φ[P (φ)→ φ(x)] A3 The property of being God-like is positive: P (G) C Possib...

Write Θ for the stable character associated to a finite dimensional representation E of a connected real reductive group G. Let M be the centralizer of a maximal torus T , and denote by ΦM (γ,Θ ) Arthur’s extension of |D M (γ)| 1 2Θ(γ) to T (R). In this paper we give a simple explicit expression for ΦM (γ,Θ ), when γ is elliptic in G.

Abstract: A new kind of entropy will be introduced which generalizes both the differential entropy and the cumulative (residual) entropy. The generalization is twofold. First, we simultaneously define the entropy for cumulative distribution functions (cdfs) and survivor functions (sfs), instead of defining it separately for densities, cdfs, or sfs. Secondly, we consider a general “entropy gener...

Let φ(n) denote Euler’s totient function, i.e., the number of positive integers < n and prime to n. We study pairs of positive integers (a0, a1) with a0 ≤ a1 such that φ(a0) = φ(a1) = (a0 + a1)/k for some integer k ≥ 1. We call these numbers φ–amicable pairs with multiplier k, analogously to Carmichael’s multiply amicable pairs for the σ–function (which sums all the divisors of n). We have comp...

Let (X,Y ) be random variables where X ∈ X is a vector of d features and Y ∈ {0, 1} represents a binary class label. A data set that contains independent observations {(xi, yi)} i=1 sampled from the joint distribution of (X,Y ) are often divided into training data {(xi, yi)}i=1 and test data {(xi, yi)} i=n+1. Based on training data, a classifier φ(·) is a mapping φ : X → {0, 1} that returns the...

Syntax Throughout this monograph, we provide the definition of the syntax of logics in a more relaxed way. Skipping the syntactic rules for brackets (which can be derived from the precedence order of the operators that will be declared in words), the above inductive definition of propositional formulae over AP can be rewritten as Φ ::= true ∣∣∣ a ∣∣∣ Φ1 ∧Φ2 ∣∣∣ ¬Φ where a ∈ AP. The above can be...

Public announcements are used in dynamic epistemic logic to model certain kinds of information change. A formula 〈ψ〉φ represents the statement that after ψ is publicly announced φ will be the case. Sometimes we want to reason about whether it is possible for φ to become true after some announcement. In order to do this an arbitrary public announcement operator ♦ can be added to an epistemic log...

Equation 1 appeared incorrectly. It should read as: () () y t y t y t y t = + − − + − − − − − µ φ µ φ µ φ φ µ 1 1 12 12 1 12 13 () + + − + − + α γ ε 1 4 13 T t a MEI t t The equations in the legend of Table 1 were also incorrect. The full model should be: () () () y t y t y t y t = + − − + − − − − − µ φ µ φ µ φ φ µ 1 1 12 12 1 12 13 + + − + − + − + − + − α α α γ γ 1 4 2 13 3 21 13 4 T t T t T t...

The KLOE detector (1) at DAΦNE, the Frascati φ-factory, has started taking data in April 1999 and a total integrated luminosity of 2.4 pb has been collected by the end of ’99, corresponding to ∼8 millions φ decays. With these data a preliminary measurement of φ radiative decays φ → ηγ, φ → πγ, φ → ηγ, φ → ππγ, φ → ππγ, φ → ηπγ and of the hadronic decay φ → πππ has been performed. The energy spe...