This study investigates prospective secondary mathematics teachers' visual representations of polynomial and rational inequalities, and graphs of exponential and logarithmic functions with GeoGebra Dynamic Softwar...
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This study investigates prospective secondary mathematics teachers' visual representations of polynomial and rational inequalities, and graphs of exponential and logarithmic functions with GeoGebra Dynamic Software. Five prospective teachers in a university in the United States participated in this research study, which was situated within a framework of productive disposition and visual representations in pre-calculus. The main result was that the role of GeoGebra as a cognitive tool fostered the research participants' productive disposition, despite recurrent mismatches between the algebraic and visualized formalisms. Moreover, participants exhibiting dynamic productive disposition seemed to understand and make better sense of the conceptual underpinnings of the mathematical content they explored in contrast to those embracing static productive disposition.
We shall investigate an application of "superexponentiation," an operation that we denote by ↑ (following [5]) and define as follows:b↑n: = b ^∧(b ^∧(... ^∧b)...) ( b>0, n = 1,2,...)where exponentia...
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We shall investigate an application of "superexponentiation," an operation that we denote by ↑ (following [5]) and define as follows:b↑n: = b ^∧(b ^∧(... ^∧b)...) ( b>0, n = 1,2,...)where exponentiation occurs n times. Superexponentiation simply continues the pattern of addition, multiplication, and exponentiation. It seems to have first appeared in the literature in [4], for the purpose of exhibiting extremely large (albeit finite) numbers (its implicit use in [7], an earlier work than [4], is shown in [5]). Superexponentiation is used in [5] and [6] to examine the logical foundation of mathematical induction. In [2] superexponentiation and its inverse operation, iteration of logarithms, are used to analyze the running time of certain algorithms. A general discussion of superexponentiation is given in [1].
The need and the demand to hide the content of written messages from prying eyes has arisen as long as writing itself has existed. Particularly, in recent times the protection of information is considered to be quite ...
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The need and the demand to hide the content of written messages from prying eyes has arisen as long as writing itself has existed. Particularly, in recent times the protection of information is considered to be quite crucial due to the increased use and development of streaming applications. Therefore, the urge to have more sophisticated, stronger and hard to break data encryption and decryption systems is increasing. In order to attain these requirements, cryptography plays an important role, where many researchers have come up with different proposals and developed algorithms that have helped out a little in ensuring the confidentiality, integrity and authentication of the given information. However, even the internet security through modern cryptography is quite complex and depends on the difficulty of certain computational problems in mathematics. Modern systems of encryption are based on complex mathematical algorithms and carry out a combination of symmetric and asymmetric key encryption schemes to secure communication. For those a significant background in algebra, number theory and geometry is required. In this paper a new cryptosystem is presented, which is based on compound commutative functions such as exponential and logarithmic functions. After the necessary theoretical deliberation in this paper, we will provide the encryption and decryption approach accompanied with relevant examples
This article discusses the definitions and properties of exponential and logarithmic functions. The treatment is based on the basic properties of real numbers, sequences and continuous functions. This treatment avoids...
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This article discusses the definitions and properties of exponential and logarithmic functions. The treatment is based on the basic properties of real numbers, sequences and continuous functions. This treatment avoids the use of definite integrals.
The problem of discrete-time and continuous-time adaptive stabilization under full-state feedback control is considered. In the discrete-time case the main result is based on a gain update law involving a step-size fu...
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The problem of discrete-time and continuous-time adaptive stabilization under full-state feedback control is considered. In the discrete-time case the main result is based on a gain update law involving a step-size function. The formulation generalizes and unifies prior results based on quadratic and logarithmic Lyapunov functions. In the continuous-time case adaptive stabilization under full-state feedback using a normalized gradient algorithm is considered and Lyapunov stability is demonstrated.
For monogenic (continuous and GA cent teaux-differentiable) functions taking values in a three-dimensional harmonic algebra with two-dimensional radical, we compute the logarithmic residue. It is shown that the logari...
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For monogenic (continuous and GA cent teaux-differentiable) functions taking values in a three-dimensional harmonic algebra with two-dimensional radical, we compute the logarithmic residue. It is shown that the logarithmic residue depends not only on the roots and singular points of a function but also on the points at which the function takes values in the radical of a harmonic algebra.
We generalize the string functions C-n,C-r(tau) associated with the coset (sl) over cap (2)(k)/u(1) to higher string functions A(n,r)(tau) and B-n,B-r(tau) associated with the coset W(k)/u(1) of the W-algebra of the l...
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We generalize the string functions C-n,C-r(tau) associated with the coset (sl) over cap (2)(k)/u(1) to higher string functions A(n,r)(tau) and B-n,B-r(tau) associated with the coset W(k)/u(1) of the W-algebra of the logarithmically extended (sl) over cap (2)(k) conformal field model with positive integer k. The higher string functions occur in decomposing W(k) characters with respect to level-k theta and Appell functions and their derivatives (the characters are neither quasiperiodic nor holomorphic, and therefore cannot decompose with respect to only theta-functions). The decomposition coefficients, to be considered "logarithmic parafermionic characters," are given by A(n,r)(tau), B-n,B-r(tau), C-n,C-r(tau), and by the triplet W(p)-algebra characters of the (p = k +2, 1) the logarithmic model. We study the properties of A(n,r) and B-n,B-r, which nontrivially generalize those of the classic string functions C-n,C-r, and evaluate the modular group representation generated from the higher level Appell functions and the associated transcendental funcion Phi.
Working in the Virasoro picture, it is argued that the logarithmic minimal models LM(p, p') = LM(p, p';1) can be extended to an infinite hierarchy of logarithmic conformal field theories LM(p, p';n) at hig...
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Working in the Virasoro picture, it is argued that the logarithmic minimal models LM(p, p') = LM(p, p';1) can be extended to an infinite hierarchy of logarithmic conformal field theories LM(p, p';n) at higher fusion levels n is an element of N. From the lattice, these theories are constructed by fusing together n x n elementary faces of the appropriate LM(p, p') models. It is further argued that all of these logarithmic theories are realized as diagonal cosets (A(1)((1)))(k) circle plus (A(1)((1)))(n)/(A(1)((1)))(k+n) where n is the integer fusion level and k = np/p'-p - 2 is a fractional level. These cosets mirror the cosets of the higher fusion level minimal models of the form M(M, M';n), but are associated with certain reducible representations. We present explicit branching rules for characters in the form of multiplication formulas arising in the logarithmic limit of the usual Goddard-Kent-Olive coset construction of the non-unitary minimal models M(M, M';n). The limiting branching functions play the role of Kac characters for the LM(p, p';n) theories.
There exist several theoretical motivations for primordial correlation functions (such as the power spectrum) to contain oscillations as a logarithmic function of comoving momentum k. While these features are commonly...
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There exist several theoretical motivations for primordial correlation functions (such as the power spectrum) to contain oscillations as a logarithmic function of comoving momentum k. While these features are commonly searched for in k space, an alternative is to use angular space, that is, search for correlations between the directional vectors of observation. We develop tools to efficiently compute the angular correlations based on a stationary phase approximation and examine several example oscillations in the primordial power spectrum, bispectrum, and trispectrum. We find that logarithmically periodic oscillations are essentially featureless and therefore difficult to detect using the standard correlator, though others might be feasible.
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