Differentiation in logical form

We introduce a logical theory of differentiation for a real-valued function on a finite dimensional real Euclidean space. A real-valued continuous function is represented by a localic approximable mapping between two semi-strong proximity lattices, representing the two stably locally compact Euclidean spaces for the domain and the range of the function. Similarly, the Clarke subgradient, equivalently the L-derivative, of a locally Lipschitz map, which is non-empty, compact and convex valued, is represented by an approximable mapping. Approximable mappings of the latter type form a bounded complete domain isomorphic with the function space of Scott continuous functions of a real variable into the domain of non-empty compact and convex subsets of the finite dimensional Euclidean space partially ordered with reverse inclusion. Corresponding to the notion of a single-tie of a locally Lipschitz function, used to derive the domain-theoretic L-derivative of the function, we introduce the dual notion of a single-knot of approximable mappings which gives rise to Lipschitzian approximable mappings. We then develop the notion of a strong single-tie and that of a strong knot leading to a Stone duality result for locally Lipschitz maps and Lipschitzian approximable mappings. The strong single-knots, in which a Lipschitzian approximable mapping belongs, are employed to define the Lipschitzian derivative of the approximable mapping. The latter is dual to the Clarke subgradient of the corresponding locally Lipschitz map defined domain-theoretically using strong single-ties. A stricter notion of strong single-knots is subsequently developed which captures approximable mappings of continuously differentiable maps providing a gradient Stone duality for these maps. Finally, we derive a calculus for Lipschitzian derivative of approximable mapping for some basic constructors and show that it is dual to the calculus satisfied by the Clarke subgradient

Hegel and the Logical Form

The concept of logical form, as influentially specified by Frege and Bolzano, is accompanied by a paradox: to capture some universal property of discourse, we must specify that property, thereby rendering it particular and thus unsuitable for the universal purpose. Thus, instead of a single form, we have rather a sequence of them, corresponding to the logics of Aristotle, Frege, Brouwer, and others. In this paper, I argue that Hegel’s conception of logical form focuses on this historical aspect of the problem. Thus, he does not create a new logical form, e.g., that of dialectical logic, as Marx, as well as Priest and others, believe, but makes the attitude towards “fixed determinations” of logic part of these determinations themselves. This corresponds to Hegel’s differentiation between three layers of logic: formal, dialectical, and speculative

Competitive strategy trends among the Malaysian wooden furniture industry: An strategic groups analysis.

The competitiveness indexes of the Malaysian furniture industry are declining regardless of the strong growth of the over two last decades. Although there are nominated factors to analyze this conflict from resource based view, other factors from competitive strategy view should be considered to attain a comprehensive image of the industry's structure in the current situation. This study developed a strategic group analysis based on Porter's competitive business level strategy to examine the industry structure and its likely association with the current situation of the industry as well strategic trends in the future. The obtained results demonstrated that: There are at least four logical strategic groups in the industry. Major groups that form main body of the industry do not pursue a distinctive competitive strategic orientation and so the industry is placed in a fragile position in terms of the competitive strategy. The industry is significantly under force from intensive competition and bargaining power of buyers. Differentiation is the most efficient and Cost leadership is the least efficient strategy in the industry. The industry significantly attempts to develop its strategic activities toward distinct strategic orientations in general and differentiation more than two other competitive strategic orientations. Hence, moving from dispassionate groups toward differentiation group will be main strategic effort in the future of the industry

Cell fate reprogramming by control of intracellular network dynamics

Identifying control strategies for biological networks is paramount for practical applications that involve reprogramming a cell's fate, such as disease therapeutics and stem cell reprogramming. Here we develop a novel network control framework that integrates the structural and functional information available for intracellular networks to predict control targets. Formulated in a logical dynamic scheme, our approach drives any initial state to the target state with 100% effectiveness and needs to be applied only transiently for the network to reach and stay in the desired state. We illustrate our method's potential to find intervention targets for cancer treatment and cell differentiation by applying it to a leukemia signaling network and to the network controlling the differentiation of helper T cells. We find that the predicted control targets are effective in a broad dynamic framework. Moreover, several of the predicted interventions are supported by experiments.Comment: 61 pages (main text, 15 pages; supporting information, 46 pages) and 12 figures (main text, 6 figures; supporting information, 6 figures). In revie

Complex Systems Analysis of Arrested Neural Cell Differentiation during Development and Analogous Cell Cycling Models in Carcinogenesis

A new approach to the modular, complex systems analysis of nonlinear dynamics of arrested neural cell Differentiation--induced cell proliferation during organismic development and the analogous cell cycling network transformations involved in carcinogenesis is proposed. Neural tissue arrested differentiation that induces cell proliferation during perturbed development and Carcinogenesis are complex processes that involve dynamically inter-connected biomolecules in the intercellular, membrane, cytosolic, nuclear and nucleolar compartments. Such 'dynamically inter-connected' biomolecules form numerous inter-related pathways referred to as 'molecular networks'. One such family of signaling pathways contains the cell cyclins. Cyclins are proteins that link several critical pro-apoptotic and other cell cycling/division components, including the tumor suppressor gene TP53 and its product, the Thomsen-Friedenreich antigen (T antigen), Rb, mdm2, c-Myc, p21, p27, Bax, Bad and Bcl-2, which play major roles in various neoplastic transformations of many tissues. The novel theoretical analysis presented here is based on recently published studies of arrested cell differentiation that normally leads to neural system formation during early developmental stages; the perturbed development may involve cyclin signaling and cell cycling responsible for rapidly induced cell proliferation without differentiation into neural cells in such experimental studies; special emphasis in this modular model is placed upon the roles of cyclins D1 and E, and does suggest novel clinical trials as well as rational therapies of cancer through re-establishment of cell cycling inhibition in metastatic cancer cells. Cyclins are proteins that are often over-expressed in cancerous cells (Dobashi et al., 2004). They may also be over-expressed in cells whose differentiation is arrested during the early stages of organismic development, leading to increased cell proliferation instead of differentiation into specialized tissues such as those forming the neural system. Cyclin-dependent kinases (CDK), their respective cyclins, and inhibitors of CDKs (CKIs) were identified as instrumental components of the cell cycle-regulating machinery. In mammalian cells the complexes of cyclins D1, D2, D3, A and E with CDKs are considered motors that drive cells to enter and pass through the “S” phase. Cell cycle regulation is a critical mechanism governing cell division and proliferation, and it is finely regulated by the interaction of cyclins with CDKs and CKIs, among other molecules (Morgan et al., 1995). A categorical and Topos framework for Łukasiewicz Algebraic Logic models of nonlinear dynamics in complex functional genomes and cell interactomes is also proposed. Łukasiewicz Algebraic Logic models of genetic networks and signaling pathways in cells are formulated in terms of nonlinear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of varying 'next-state' functions is extended in a Łukasiewicz-Topos with an n-valued Łukasiewicz Algebraic Logic subobject classifier description that represents non-random and nonlinear network activities as well as their transformations in developmental processes and carcinogenesis. Important aspects of Cell Cycling, the Control of Cell Division,and the Neoplastic Transformation in Carcinogenesis are being considered and subjected to algebraic-logico- relational, and computer-aided investigations. The essential roles of various levels of c-Myc, p27 quasi-complete inhibition/blocking, TP53 and/or p53 inactivation, as well as the perpetual hTERT activation of Telomerase biosynthesis are pointed out as key conditions for Malignant Cell transformations and partial re-differentiation leading to various types of cancer such as lung, breast,skin, prostate and colon. Rational Clinical trials, Individualized Medicine and the potential for optimized Radio-, Chemo-, Gene-, and Immuno- therapies of Cancers are suggested on the basis of integrated complex systems biology modeling of oncogenesis, coupled with extensive genomic/proteomic and interactomic High-throughput/high-sensitivity measurements of identified, sorted cell lines that are being isolated from malignant tumors of patients undergoing clinical trials with adjuvant signaling drug therapies. The implications of the cyclin model for abnormal neural development during early development are being considered in this model that may lead to explanations of subsequent cognitive changes associated with abnormal neural cell differentiation in environmentally-affected embryos. This new model may also be relevant to detecting the onset of senescing neuron transformations in Alzheimer's and related diseases of the human brain in ageing populations at risk

A Method to Identify and Analyze Biological Programs through Automated Reasoning.

Predictive biology is elusive because rigorous, data-constrained, mechanistic models of complex biological systems are difficult to derive and validate. Current approaches tend to construct and examine static interaction network models, which are descriptively rich but often lack explanatory and predictive power, or dynamic models that can be simulated to reproduce known behavior. However, in such approaches implicit assumptions are introduced as typically only one mechanism is considered, and exhaustively investigating all scenarios is impractical using simulation. To address these limitations, we present a methodology based on automated formal reasoning, which permits the synthesis and analysis of the complete set of logical models consistent with experimental observations. We test hypotheses against all candidate models, and remove the need for simulation by characterizing and simultaneously analyzing all mechanistic explanations of observed behavior. Our methodology transforms knowledge of complex biological processes from sets of possible interactions and experimental observations to precise, predictive biological programs governing cell function

Cross-Product Extensions of the Gene Ontology

The Gene Ontology is being normalized and extended to include computable logical definitions. These definitions are partitioned into mutually exclusive cross-product sets, many of which reference other OBO Foundry ontologies. The results can be used to reason over the ontology, and to make cross-ontology queries