Nearly General Septic Functional Equation

If a mapping can be expressed by sum of a septic mapping, a sextic mapping, a quintic mapping, a quartic mapping, a cubic mapping, a quadratic mapping, an additive mapping, and a constant mapping, we say that it is a general septic mapping. A functional equation is said to be a general septic functional equation provided that each solution of that equation is a general septic mapping. In fact, there are a lot of ways to show the stability of functional equations, but by using the method of G a ˘ vruta, we examine the stability of general septic functional equation ∑ i = 0 8 C 8 i − 1 8 − i f x + i − 4 y = 0 which considered. The method of G a ˘ vruta as just mentioned was given in the reference Gavruta (1994).

Machine Learning Human Rights and Wrongs: How the Successes and Failures of Supervised Learning Algorithms Can Inform the Debate About Information Effects

There is an ongoing debate about whether human rights standards have changed over the last 30 years. The evidence for or against this shift relies upon indicators created by human coders reading the texts of human rights reports. To help resolve this debate, we suggest translating the question of changing standards into a supervised learning problem. From this perspective, the application of consistent standards over time implies a time-constant mapping from the textual features in reports to the human coded scores. Alternatively, if the meaning of abuses have evolved over time, then the same textual features will be labeled with different numerical scores at distinct times. Of course, while the mapping from natural language to numerical human rights score is a highly complicated function, we show that these two distinct data generation processes imply divergent overall patterns of accuracy when we train a wide variety of algorithms on older versus newer sets of observations to learn how to automatically label texts with scores. Our results are consistent with the expectation that standards of human rights have changed over time.

On type-2 m-topological spaces

In the present paper, we define a notion of an m2-topological space by introducing a count of openness of a multiset (mset in short) and study the properties of m2-subspaces, mgp-maps etc. Decomposition theorems involving m-topologies and m2-topologies are established. The behaviour of the functional image and functional preimage of an m2-topologies, the continuity of the identity mapping and a constant mapping in m2-topologies are also examined.

Mappings of finite distortion: a new proof for discreteness and openness

We give a new and elementary proof of the known result: a non-constant mapping of finite distortion f : Ω ⊂ ℝn → ℝn is discrete and open, provided that its distortion function if n = 2 and that for some p > n − 1 if n ≥ 3.

IMAGING AND MAPPING OF INDIVIDUAL NANOPARTICLES AND NANOTUBES BY SELECTIVE NANOVOLUME RAMAN SPECTROSCOPY

In this experiment, we report on a new procedure to analyze nanoparticles, like individual single-walled carbon nanotubes (SWNT), based on an improved use of a confocally selective nanovolume Raman spectroscopy set-up coupled to a refined signal treatment. This concept is based on actually available imaging microspectrometry systems for working in the near-field domain combined with a stigmatic solid immersion lens. The combination of different techniques with modified and configured equipment entitles us to expand the functionality toward a three-dimensional mapping of Raman and photoluminescence intensity with a possible discrimination in polarization as well as photoluminescence decay time constant mapping. This technique permits to further improve the resolution in both the vertical and lateral dimensions. To acquire a spectrally selected SWNT, we registered the spatial distribution of the emitted photons in x, y, z vectors with an unprecedented accuracy and were able to resolve details in the specimen with a near-field resolution. In this experiment, we report what we believe to be the first evidence of selective nanovolume Raman spectroscopy of individual nanoparticles.

Local properties of maps of the ball

Letfbe an essential map ofSn−1into itself (i.e.,fis not homotopic to a constant mapping) admitting an extension mapping the closed unit ballB¯nintoℝn. Then, for every interior pointyofBn, there exists a pointxinf−1(y)such that the image of no neighborhood ofxis contained in a coordinate half space withyon its boundary. Under additional conditions, the image of a neighborhood ofxcovers a neighborhood ofy. Differential versions are valid for quasianalytic functions. These results are motivated by game-theoretic considerations.