In this work, we explore modeling of wormholes in framework of f(R, T) gravity with the functional form f(R,T)=R+αR2+λT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(R, T)= R+\alpha R^2 +\lambda T$$\end{document}, where R and T are the Ricci scalar and trace of energy-momentum tensor respectively, α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} and λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document} are arbitrary constants. Using the equation of state (EoS) pr=ωρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} (...) \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_r=\omega \rho$$\end{document} and shape function b(r)=r0arar0,(0shrink)

Plato provocatively characterizes truth $$ in terms of harmony $$ at various points throughout his dialogues. While limited attention has been directed toward the role of musical concepts in Plato's general cosmology, not any attention has been directed toward how musical concepts function in relation to Plato's characterization of truth. In fact, this issue has had little occasion for consideration. Almost every contemporary translator empties terms such as $\grave\alpha\rho\mu o\nu\acute\iota\alpha,$ when co-incidental with $\acute\alpha\lambda\acute\eta\theta\varepsilon\iota\alpha,$ of their musical content. As a consequence, (...) I argue, Plato's characterization of truth is made readily amenable to contemporary conceptions of truth-not necessarily a good thing since it robs us of the original richness of Plato's thought. The dissertation explores how retaining the aural meaning latent within $\grave\alpha\rho\mu o\nu\acute\iota\alpha$ and other terms of musical import serves to elucidate the rich complexion of Plato's notion of $\acute\alpha\lambda\acute\eta\theta\varepsilon\iota\alpha.$ Ultimately, I show how Plato's use of musical terminology to characterize truth informs other features of his philosophical outlook, illustrate the cohesiveness and consistency of key texts believed problematic by other commentators, and underscore significant differences between Plato's $\acute\alpha\lambda\acute\eta\theta\varepsilon\iota\alpha$ and ideas of truth prevalent today. ;For Plato, I argue $\acute\alpha\lambda\acute\eta\theta\varepsilon\iota\alpha$ bears a meaning different from its primitive sense of oracular unconcealment and its later sense of correctness $,$ and a meaning also different from other terminological alternatives $.$ Truth, accordingly, turns out for Plato to be neither mysterious and groundless nor a matter of simple correspondence to an objective reality. The value of Plato's conception of truth as harmony consists in its ability to avoid the polarity of contemporary conceptions of truth as either objectively determinate or subjectively determinate . Truth is neither arbitrarily imposed upon the world by us, nor artificially imposed upon us by an antecedently structured and independent reality. The dissertation concludes by articulating reasons for favoring Plato's theory of truth, while noting a few underlying assumptions likely problematic for some contemporary philosophers. (shrink)

In this second part of “Negative Größen bei Diophant?” we start, as announced, by giving 33 places where Diophantus uses negative quantities as intermediate results; they appear as differences a − b of positive rational numbers, the subtrahend b being bigger than the minuend a; they each represent the (negative) basis $(\pi\lambda\varepsilon\upsilon\rho\acute{\alpha})$ of a square number $(\tau\varepsilon\tau\rho\acute{\alpha}\gamma\omega\nu o \zeta)$ , which is afterwards computed by the formula (a - b)2 = a 2 + b 2 - 2ab. Finally, we report (...) how the topic “Diophantus and the negative numbers” has been dealt with by translators and commentators from Maximus Planudes onwards. (shrink)

My dissertation is concerned with Epicurus' attempt to reconcile libertarianism and atomism. I begin by offering my solution to 'the problem of the swerve,' arguing that Lucretius is claiming that swerves cause volitions 'from the bottom up' and that the attempts of scholars to construct a better position for Epicurus to have held were doomed to fail, since this is the only position open to the libertarian atomist. ;I also examine the swerve's role in cosmogony, arguing that 'the cosmogonic argument' (...) for the swerve is an afterthought, 'the libertarian argument' being Epicurus' chief argument for the swerve. ;I discuss the physics of the swerve as a minimal, oblique deviation of an atom from its course. And I explore its implications: atoms move at a speed greater than one minimal unit of distance per minimal unit of time, and there are no minimal units of space. ;Returning to the question of the swerve's role in action, I argue that a typical action involves a series of volitions each caused by a plurality of swerves by atoms of the mind's 'fourth nature.' And I argue that one's power to move at will is directly sensed, though also known through one's innate $\pi\rho\acute{o}\lambda\eta\psi\iota\varsigma$ of one's freedom. ;In the second half of my dissertation, I turn to the question of responsibility. I examine Epicurus' remarks about 'what is due to us' . I examine Epicurus' argument that Democriteans are self-refuting in espousing determinism and 'eliminative materialism.' I explain that Epicurus rejects only Democritus' ontology, not his epistemology . I examine Epicurus' discussion in On Nature of our 'notion of responsibility' , our responsibility as $\alpha\pi o\gamma\varepsilon\gamma\varepsilon\nu\nu\eta\mu\acute \varepsilon\nu\alpha$ , and our responsibility for our thoughts of the $\tau\acute\varepsilon\lambda o\varsigma$ and 'the criterion' . I explain what Epicurus means by the $\tau\acute\varepsilon\lambda o\varsigma$: pleasure, "katastematic" and "kinetic" . And finally, I argue that Epicurus fails to meet the challenge of 'psychological determinism'. (shrink)

We construct the Extended Relativity Theory in Born-Clifford-Phase spaces with an upper R and lower length λ scales (infrared/ultraviolet cutoff). The invariance symmetry leads naturally to the real Clifford algebra Cl (2, 6, R) and complexified Clifford Cl C (4) algebra related to Twistors. A unified theory of all Noncommutative branes in Clifford-spaces is developed based on the Moyal-Yang star product deformation quantization whose deformation parameter involves the lower/upper scale $$(\hbar \lambda / R)$$. Previous work led us to show from (...) first principles why the observed value of the vacuum energy density (cosmological constant) is given by a geometric mean relationship $$\rho \sim L_{\rm Planck}^{-2}R^{-2} = L_{P}^{-4} (L_{\rm Planck} / R)^{2} \sim 10^{-122}M_{\rm Planck}^4$$, and can be obtained when the infrared scale R is set to be of the order of the present value of the Hubble radius. We proceed with an extensive review of Smith’s 8D model based on the Clifford algebra Cl (1, 7) that reproduces at low energies the physics of the Standard Model and Gravity, including the derivation of all the coupling constants, particle masses, mixing angles,....with high precision. Geometric actions are presented like the Clifford-Space extension of Maxwell’s Electrodynamics, and Brandt’s action related to the 8D spacetime tangent-bundle involving coordinates and velocities (Finsler geometries). Finally we outline the reasons why a Clifford-Space Geometric Unification of all forces is a very reasonable avenue to consider and propose an Einstein-Hilbert type action in Clifford-Phase spaces (associated with the 8D Phase space) as a Unified Field theory action candidate that should reproduce the physics of the Standard Model plus Gravity in the low energy limit. (shrink)

Following Dirac's generalized canonical formalism, we develop a quantization scheme for theN-dimensional system described by the Lagrangian $L_0 (\dot y,y) = \frac{1}{2}h_{ij} (y)\dot y^i \dot y^j + b_i (y)\dot y^i - w(y)$ which is supposed to be invariant under the gauge transformation $y^i \to y\prime ^i = y^i + (\rho ^i _\alpha + \sigma ^i _{\alpha j} \dot y^j )\delta \Lambda ^\alpha + \tau ^i _\alpha \delta \dot \Lambda ^\alpha$ . The gauge invariance necessarily implies that the Lagrangian is singular. (...) The identities imposed by the gauge invariance are enumerated and reduced to simpler forms. There are primary and secondary constraints, both of which are of first class. The reduced identities are solved explicitly for the case where the secondary constraints constitute the generators of the groupSO(M), and thus an explicit expression for the manifestly gauge-invariant Lagrangian is obtained. By fixing the gauge appropriately, the unphysical variables are eliminated and a quantization is achieved using only physical variables. Our formulation is covariant under an arbitrary point transformation of physical variables. The problem of formulating a quantum action principle is also commented on briefly. (shrink)