On a Conditional Cauchy-type Functional Equation Involving Powers

We solve the functional equation f(x^m + y) = f(x)^m + f(y) in the realm of polynomials with integer coefficients.

Powers and Logarithms

There are applied power mappings in algebras with logarithms induced by a given linear operator D in order to study particular properties of powers of logarithms. Main results of this paper will be concerned with the case when an algebra under consideration is commutative and has a unit and the operator D satisfies the Leibniz condition, i.e. D(xy) = xDy + yDx for x, y ∈ dom D. Note that in the Number Theory there are well-known several formulae expressed by means of some combinations of powers of logarithmic and antilogarithmic mappings or powers of logarithms and antilogarithms (cf. for instance, the survey of Schinzel S[1].

Fractional Derivatives and Fractional Powers as Tools in Understanding Wentzell Boundary Value Problems for Pseudo-Differential Operators Generating Markov Processes

Mathematics Subject Classification: 26A33, 31C25, 35S99, 47D07.

Fractional Powers of Almost Non-Negative Operators

Mathematics Subject Classification: Primary 47A60, 47D06.

Bounds for Fractional Powers of Operators in a Hilbert Space and Constants in Moment Inequalities

Mathematics Subject Classification: 47A56, 47A57,47A63

An Analysis on the Decline in the Authoritative Powers of the Papacy and its Application to Challenges to Authoritative Powers Faced By the Permanent Membership of the UN Security Council

The organizational authority of the Papacy in the Roman Catholic Church and the permanent membership of the UN Security Council are unique from institutions that are commonly compared with the UN, like the Concert of Europe and the League of Nations, in that these institutional organs possessed strong authoritative and veto powers. Both organs also owe their strong authority during their founding to a need for stability: The Papacy after the crippling of Western Roman Empire and the P-5 to deal with the insecurities of the post-WWII world. While the P-5 still possesses similar authoritative powers within the Council as it did after WWII, the historical authoritative powers of the Papacy within the Church was debilitated to such a degree that by the time of the Reformation in Europe, condemnations of practices within the Church itself were not effective. This paper will analyze major challenges to the authoritative powers of the Papacy, from the crowning of Charlemagne to the beginning of the Reformation, and compare the analysis to challenges affecting the authoritative powers of the P-5 since its creation. From research conducted thus far, I hypothesize that common themes affecting the authoritative powers of the P-5 and the Papacy would include: major changes in the institutions organization (i.e. the Avignon Papacy and Japan’s bid to become a permanent member); the decline in power of actors supporting the institutional organ (i.e. the Holy Roman Empire and the P-5 members); and ideological clashes affecting the institution’s normative power (i.e. the Great Western Schism and Cold War politics).

An Analysis on the Decline in the Authoritative Powers of the Papacy and its Application to Challenges to Authoritative Powers Faced By the Permanent Membership of the UN Security Council

The organizational authority of the Papacy in the Roman Catholic Church and the permanent membership of the UN Security Council are unique from institutions that are commonly compared with the UN, like the Concert of Europe and the League of Nations, in that these institutional organs possessed strong authoritative and veto powers. Both organs also owe their strong authority during their founding to a need for stability: The Papacy after the crippling of Western Roman Empire and the P-5 to deal with the insecurities of the post-WWII world. While the P-5 still possesses similar authoritative powers within the Council as it did after WWII, the historical authoritative powers of the Papacy within the Church was debilitated to such a degree that by the time of the Reformation in Europe, condemnations of practices within the Church itself were not effective. This paper will analyze major challenges to the authoritative powers of the Papacy, from the crowning of Charlemagne to the beginning of the Reformation, and compare the analysis to challenges affecting the authoritative powers of the P-5 since its creation. From research conducted thus far, I hypothesize that common themes affecting the authoritative powers of the P-5 and the Papacy would include: major changes in the institutions organization (i.e. the Avignon Papacy and Japan’s bid to become a permanent member); the decline in power of actors supporting the institutional organ (i.e. the Holy Roman Empire and the P-5 members); and ideological clashes affecting the institution’s normative power (i.e. the Great Western Schism and Cold War politics).

An investigation of lower limb joint powers as a predictive measure of countermovement vertical jump height.

Measuring and tracking athletic performance is crucial to an athlete’s development and the countermovement vertical jump is often used to measure athletic performance, particularly lower limb power. The linear power developed in the lower limb is estimated through jump height. However, the relationship between angular power, produced by the joints of the lower limb, and jump height is not well understood. This study examined the contributions of the kinetic value of angular power, and its kinematic component, angular velocity, of the lower limb joints to jump height in the countermovement vertical jump. Kinematic and kinetic data were gathered from twenty varsity-level basketball and volleyball athletes as they performed six maximal effort jumps in four arm swing conditions: no-arm involvement, single-non-dominant arm swing, single-dominant arm swing, and two-arm swing. The displacement of the whole body centre of mass, peak joint powers, peak angular velocity, and locations of the peaks as a percentage of the jump’s takeoff period, were computed. Linear regressions assessed the relationship of the variables to jump height. Results demonstrated that knee peak power (p = 0.001, ß = 0.363, r = 0.363), its location within takeoff period (p = 0.023, ß = -0.256, r = 0.256), and peak knee peak angular velocity (p = 0.005, ß = 0.310, r = 0.310) were moderately linked to increased jump height. Additionally, the location, within the takeoff period, of the peak angular velocities of the hip (p = 0.003, ß = -0.318, r = 0.419) and ankle (p = 0.011, ß = 0.270, r = 0.419) were positively linked to jump height. These results highlight the importance of training the velocity and timing of joint motion beyond traditional power training protocols as well as the importance of further investigation into appropriate testing protocol that is sensitive to the contributions by individual joints in maximal effort jumping.