We define a first-order extension LK(CP) of the cutting planes proof system CP as the first-order sequent calculus LK whose atomic formulas are CP-inequalities ∑ i a i · x i ≥ b (x i 's variables, a i 's and b constants). We prove an interpolation theorem for LK(CP) yielding as a corollary a conditional lower bound for LK(CP)-proofs. For a subsystem R(CP) of LK(CP), essentially resolution working with clauses formed by CP- inequalities, we prove a monotone (...) interpolation theorem obtaining thus an unconditional lower bound (depending on the maximum size of coefficients in proofs and on the maximum number of CP-inequalities in clauses). We also give an interpolation theorem for polynomial calculus working with sparse polynomials. The proof relies on a universal interpolation theorem for semantic derivations [16, Theorem 5.1]. LK(CP) can be viewed as a two-sorted first-order theory of Z considered itself as a discretely ordered Z-module. One sort of variables are module elements, another sort are scalars. The quantification is allowed only over the former sort. We shall give a construction of a theory LK(M) for any discretely ordered module M (e.g., LK(Z) extends LK(CP)). The interpolation theorem generalizes to these theories obtained from discretely ordered Z-modules. We shall also discuss a connection to quantifier elimination for such theories. We formulate a communication complexity problem whose (suitable) solution would allow to improve the monotone interpolation theorem and the lower bound for R(CP). (shrink)

We introduce a notion of simplicity for types in discretely ordered first order structures. We prove that all the structure on the locus of a simple type is induced exclusively by the ordering relation. As an application we determine all possible expansions of satisfying CB = 1.

Linear arithmetics are extensions of Presburger arithmetic () by one or more unary functions, each intended as multiplication by a fixed element (scalar), and containing the full induction schemes for their respective languages. In this paper, we construct a model of the 2‐linear arithmetic (linear arithmetic with two scalars) in which an infinitely long initial segment of “Peano multiplication” on is ‐definable. This shows, in particular, that is not model complete in contrast to theories and that are known to satisfy (...) quantifier elimination up to disjunctions of primitive positive formulas. As an application, we show that, as a discretely ordered module over the discretely ordered ring generated by the two scalars, does not have the NIP, answering negatively a question of Chernikov and Hils. (shrink)

Three classes of decidable discrete linear orders with varying degrees of effectiveness are investigated. We consider how a classical order type may lie in relation to these three classes, and we characterize by their order types elements of these classes that have effective nontrivial self-embeddings.

Advocates of dynamic systems have suggested that higher mental processes are based on continuous representations. In order to evaluate this claim, we first define the concept of representation, and rigorously distinguish between discrete representations and continuous representations. We also explore two important bases of representational content. Then, we present seven arguments that discrete representations are necessary for any system that must discriminate between two or more states. It follows that higher mental processes require discrete representations. We (...) also argue that discrete representations are more influenced by conceptual role than continuous representations. We end by arguing that the presence of discrete representations in cognitive systems entails that computationalism (i.e., the view that the mind is a computational device) is true, and that cognitive science should embrace representational pluralism. (shrink)

H.L.A. Hart’s lost and found essay ‘Discretion’ has provided new insight into the issue of how legal systems can cope with indeterminacy in the law. The so-called ‘open texture’ of law calls for the exercise of judicial discretion, which, I argue, renders judges susceptible to the problem of dirty hands. To show this, I frame the problem as being open to an array of appropriate emotional responses, namely, various senses of guilt. With these responses in mind, I revise an example (...) from Michael Walzer’s original analysis in a way that highlights purely personal sacrifices in solutions to dirty hands situations. I then turn to an account of moral emotions in legal decision-making and show how judges—in failing to advance all interests—might be left with a unique sense of guilt. With an application of this account to Hart’s legal positivism, it can be seen that a judge’s hands are often dirtied in resolving borderline cases. If discretion leaves judges in situations where they must do wrong in order to do right, Hart’s endorsement of a closure view of wrongdoing will lead to difficulties in how he can explain the presence of moral remainders in jurisprudence. (shrink)

We examine a case in which non-computable behavior in a model is revealed by computer simulation. This is possible due to differing notions of computability for sets in a continuous space. The argument originally given for the validity of the simulation involves a simpler simulation of the simulation, still further simulations thereof, and a universality conjecture. There are difficulties with that argument, but there are other, heuristic arguments supporting the qualitative results. It is urged, using this example, that absolute validation, (...) while highly desirable, is overvalued. Simulations also provide valuable insights that we cannot yet (if ever) prove. (shrink)

A general metatheorem is proved which reduces a wide class of statements about continuous time stochastic processes to statements about discrete time processes. We introduce a strong language for stochastic processes, and a concept of forcing for sequences of discrete time processes. The main theorem states that a sentence in the language is true if and only if it is forced. Although the stochastic process case is emphasized in order to motivate the results, they apply to a (...) wider class of random variables. At the end of the paper we illustrate how the theorem can be used with three applications involving submartingales. (shrink)

Discrete weakly o-minimal structures, although not so stimulating as their dense counterparts, do exhibit a certain wealth of examples and pathologies. For instance they lack prime models and monotonicity for definable functions, and are not preserved by elementary equivalence. First we exhibit these features. Then we consider a countable theory of weakly o-minimal structures with infinite definable discrete subsets and we study the Boolean algebra of definable sets of its countable models.

A concise yet rigorous introduction to logic and discrete mathematics. This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the (...) semantics of classical logic as well as practical applications through the easy to understand and use deductive systems of Semantic Tableaux and Resolution. The chapters on set theory, number theory, combinatorics and graph theory combine the necessary minimum of theory with numerous examples and selected applications. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in the accompanying solutions manual. Key Features: Suitable for a variety of courses for students in both Mathematics and Computer Science. Extensive, in-depth coverage of classical logic, combined with a solid exposition of a selection of the most important fields of discrete mathematics Concise, clear and uncluttered presentation with numerous examples. Covers some applications including cryptographic systems, discrete probability and network algorithms. Logic and Discrete Mathematics: A Concise Introduction is aimed mainly at undergraduate courses for students in mathematics and computer science, but the book will also be a valuable resource for graduate modules and for self-study. (shrink)

Aggregation of variables allows to approximate a large scale dynamical system (the micro-system) involving many variables into a reduced system (the macro-system) described by a few number of global variables. Approximate aggregation can be performed when different time scales are involved in the dynamics of the micro-system. Perturbation methods enable to approximate the large micro-system by a macro-system going on at a slow time scale. Aggregation has been performed for systems of ordinary differential equations in which time is a continuous (...) variable. In this contribution, we extend aggregation methods to time-discrete models of population dynamics. Time discrete micro-models with two time scales are presented. We use perturbation methods to obtain a slow macro-model. The asymptotic behaviours of the micro and macro-systems are characterized by the main eigenvalues and the associated eigenvectors. We compare the asymptotic behaviours of both systems which are shown to be similar to a certain order. (shrink)

In recent decades, plural logic has established itself as a well-respected member of the extensions of first-order classical logic. In the present paper, I draw attention to the fact that among the examples that are commonly given in order to motivate the need for this new logical system, there are some in which the elements of the plurality in question are internally singularized (e.g. ‘Whitehead and Russell wrote Principia Mathematica’), while in others they are not (e.g. ‘Some philosophers (...) wrote Principia Mathematica’). Then, building on previous work, I point to a subsystem of plural logic in which inferences concerning examples of the first type can be adequately dealt with. I notice that such a subsystem (here called ‘discrete plural logic’) is in reality a mere variant of first-order logic as standardly formulated, and highlight the fact that it is axiomatizable while full plural logic is not. Finally, I urge that greater attention be paid to discrete plural logic and that discrete plurals are not used in order to motivate the introduction of full-fledged plural logic—or, at least, not without remarking that they can also be adequately dealt with in a considerably simpler system. (shrink)

Intelligible speakers achieve specific vocal tract constrictions in rapid sequence. These constrictions are associated in theory with speech motor goals. Adult-focused models of speech production assume that discrete phonological representations, sequenced into word-length plans for output, define these goals. This assumption introduces a serial order problem for speech. It is also at odds with children's speech. In particular, child phonology and timing control suggest holistic speech plans, and so the hypothesis of whole word production. This hypothesis solves the (...) serial order problem by avoiding it. When the same solution is applied to adult speech the problem becomes how to explain the development of highly intelligible speech. This is the problem addressed here. A modeling approach is used to demonstrate how perceptual-motor units of production emerge over developmental time with the perceptual-motor integration of holistic speech plans that are also phonological representations; the specific argument is that perceptual-motor units are a product of trajectories (nearly) crossing in motor space. The model, which focuses on the integration process, defines the perceptual-motor map as a set of linked pairs of experienced perceptual and motor trajectories. The trajectories are time-based excursions through speaker-defined perceptual and motor spaces. By hypothesis, junctures appear where motor trajectories near or overlap one another in motor space when the shared (or extremely similar) articulatory configurations in these regions are exploited to combine perceptually-linked motor paths along different trajectories. Junctures form in clusters in motor space. These clusters, along with their corresponding (linked) perceptual points, represent perceptual-motor units of production, albeit at the level of speech motor control only. The units serve as pivots in motor space during speaking; they are points of transition from one motor trajectory to another along perceptually-linked paths that are selected to produce best approximations of whole word targets. (shrink)

We prove that the classical theory with a discrete time (chronon) is a particular case of a more general theory in which spinning particles are associated with generalized Lagrangians containing time-derivatives of any order (a theory that has been called “Non-Newtonian Mechanics”). As a consequence, we get, for instance, a classical kinematical derivation of Hamiltonian and spin vector for the mentioned chronon theory (e.g., in Caldirola et al.’s formulation). Namely, we show that the extension of classical mechanics obtained (...) by the introduction of an elementary time-interval does actually entail the arising of an intrinsic angular momentum; so that it may constitute a possible alternative to string theory in order to account for the internal degrees of freedom of the microsystems. (shrink)

Fix an integer $r \ge 3$. We consider metric spaces on n points such that the distance between any two points lies in $\left\{ {1, \ldots,r} \right\}$. Our main result describes their approximate structure for large n. As a consequence, we show that the number of these metric spaces is $\left\lceil {{{r + 1} \over 2}} \right\rceil ^{\left + o\left}.$Related results in the continuous setting have recently been proved by Kozma, Meyerovitch, Peled, and Samotij [34]. When r is even, our (...) structural characterization is more precise and implies that almost all such metric spaces have all distances at least $r/2$. As an easy consequence, when r is even, we improve the error term above from $o\left$ to $o\left$, and also show a labeled first-order 0-1 law in the language ${\cal L}_r $, consisting of r binary relations, one for each element of $[r]$. In particular, we show the almost sure theory T is the theory of the Fraïssé limit of the class of all finite simple complete edge-colored graphs with edge colors in $\left\{ {r/2, \ldots,r} \right\}$.Our work can be viewed as an extension of a long line of research in extremal combinatorics to the colored setting, as well as an addition to the collection of known structures that admit logical 0-1 laws. (shrink)

Aesthetic investigations of computation are stuck in an impasse, caused by the difficulty of accounting for the ontological discrepancy between the continuity of sensation and the discreteness of digital technology. This article proposes a theoretical position intended to overcome that deadlock. It highlights how an ontological focus on continuity has entered media studies via readings of Deleuze, which attempt to build a ‘digital aisthesis’ by ascribing a ‘virtuality’ to computation. This underpins, in part, the affective turn in digital theory. In (...) contrast to such positions, this article argues for a reconceptualization of formal abstraction in computation, in order to find, within the discreteness of computational formalisms, an indeterminacy that would make computing aesthetic qua inherently generative. This indeterminacy, it is argued here, can be found by reconsidering, philosophically, Turing’s notion of ‘incomputability’. (shrink)

AUTHOR: STAN GUDDER (John Evans Professor of Mathematical Physics, University of Denver, USA) -- -/- We consider a discrete scalar, quantum field theory based on a cubic 4-dimensional lattice. We mainly investigate a discrete scattering operator S(x0,r) where x0 and r are positive integers representing time and maximal total energy, respectively. The operator S(x0,r) is used to define transition amplitudes which are then employed to compute transition probabilities. These probabilities are conditioned on the time-energy (x0,r). In order (...) to maintain total unit probability, the transition probabilities need to be reconditioned at each (x0,r). This is roughly analogous to renormalization in standard quantum field theory, except no infinities or singularities are involved. We illustrate this theory with a simple scattering experiment involving a common interaction Hamiltonian. We briefly mention how discreteness of space-time might be tested astronomically. Moreover, these tests may explain the existence of dark energy and dark matter. (shrink)

A theory of higher-order vagueness for partially-defined, context-sensitive predicates like is blue is offered. According to the theory, the predicate is determinately blue means roughly is an object o such that the claim that o is blue is a necessary consequence of the rules of the language plus the underlying non-linguistic facts in the world. Because the question of which rules count as rules of the language is itself vague, the predicate is determinately blue is both vague and partial (...) in a sense analogous (though not completely identical) to the sense in which is blue is vague and partial. However, higher-order vagueness stops there. Although the anti-extension of is determinately blue differs from that of is blue, further iterations of determinately have no semantic effect. The paper closes with a discussion of the need to posit sharp and precise lines dividing the range of potential application of a vague predicate into semantically distinct categories. It is argued that this is less of a problem for the present theory than for many others, because the categories into which the range is divided by the present theory -- like the category of counting as blue because the rules of the language offer speakers no discretion, as opposed to the category of counting as blue because speakers have invoked an absolute minimum of discretion in adjusting the contextually sensitive boundaries of the predicate -- are not those that the linguistic competence of ordinary speakers equips them to make highly accurate and reliable judgments about. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:First Order Relationality and Its Implications:A Response to David ElsteinRoger T. Ames (bio)David Elstein has asked a series of important questions about Human Becomings that provide me with an opportunity to try to bring the argument of the book into clearer focus. Let me begin by thanking David for his always generous and intelligent reflection on not only my new monograph [End Page 181] but also on Henry (...) Rosemont's and my best efforts to expand upon the idea of "role ethics" as a productive way to read Confucian ethics.David asks: Is relationality truly a part of original Confucian thought? Does relationality rule out any notion of essential human nature, and does it necessarily imply or entail role ethics? Is relationality true or merely a useful way of thinking (if these can be distinguished)? Do we know that it's useful?I think the answer to these questions turns on two very different ways of understanding relationality, namely the doctrines of first and second order relationality. The term 'ontology' as "the science of being per se or in itself" gives us a doctrine of external, second order relations. Ontology was introduced into our philosophical vocabulary as a new term for an old way of thinking at the beginning of the seventeenth century by two German philosophers.1 For the classical Greeks, "the science of being per se" (to on he on) was posited as their "first philosophy" with all of its implications for metaphysics, epistemology, logic, ethics, philosophy of religion, and so on. Ontology is that branch of metaphysics that seeks to classify and explain the things that exist, with its underlying assumption that there are unchanging substances or essences internal to things that are available to us to classify them as this and not that. Ontology privileges "being per se" with its categorical language and its "substance" and "attribute" dualism, giving us a grammar of substances as property-bearers and properties that are borne, respectively.Such ontological thinking animates Plato's pursuit of formal, "real" definitions in his quest for certainty (that is, definitions not of words but of what really is), and underlies Aristotle's taxonomical science of knowing "what is what." For these classical Greek philosophers, only what is real and is thus true can be the proper object of knowledge, giving us a logic of the changeless. Corollary to ontology is causal thinking, formal definitions, and, important for David's question, a doctrine of external, second order relationality in which relations conjoin discrete and self-sufficient things, such as human "beings." As the contemporary philosopher Zhao Tingyang 趙汀陽 avers, this kind of substance ontology defining the real things that constitute the content of an orderly and structured cosmosprovides a "dictionary" kind of explanation of the world, seeking to set up an accurate understanding of the limits of all things. In simple terms, it determines "what is what," and all concepts are footnotes to "being" or "is."2In the Book of Changes, we find a vocabulary making explicit cosmological assumptions that stand in stark contrast to this substance ontology and that brings with it a doctrine of first order, constitutive relations. This alternative "first philosophy" has prompted me to create the neologism "zoetology," with the Greek zoe- "life" and -logia "discourse," as my own new term for an old way of thinking that we might translate into modern Chinese as shengshenglun 生生論: "the art of living." The starting point in this zoetological cosmology, then, is that nothing does anything by [End Page 182] itself; association is a fact. Since the very nature of life is associative and transactional, the vocabulary appealed to in defining Confucian cosmology is irreducibly dyadic and collateral: always multiple, never one.And further, it is the nature of life itself that it seeks to optimize the available conditions for its continuing growth, where such growth has a persistent existential and thus intentional aspect. "Life" is the productive correlation between organism and environment that gives rise to correlative, relational, ecological thinking that many of our best sinologists (Marcel Granet, David Keightley, Tang Junyi, K. C. Chang, A. C. Graham, and many others) would ascribe to the earliest semblances of Chinese... (shrink)

In today's world, businesses are involved in several different initiatives to gain sustainable performance, which can discourse the expectations and demands of society. Emerging economics faces numerous challenges in terms of social, relational, governance, and financial, which made it necessary for firms to perform responsibly in order to make positive contributions toward sustainability. Therefore, this study based on upper-echelon theory constructs a comprehensive framework on responsible leadership, corporate social responsibility, and managerial discretion to provide the guideline for business sustainability. (...) To fulfill the study purpose, an empirical analysis was performed on 382 Pakistani respondents' data by using PLS-SEM through SmartPLS. Outcomes of the study indicated that responsible leadership entails enormous importance in corporations to influence their impact on society, environment, and economy, which leads to business sustainability. Moreover, the study provides insights into that more responsible and experienced leaders are better able to identify the concerns of their firm's stakeholders in the manners of relational, governing, and orientational perspectives, which help them in satisfying the stakeholders' and society's demands. Hence, responsible leadership enables the firm to better strive for sustainability by applying better CSR activities and directing the decisions by maintaining the compliance of the managers. Furthermore, the current study concluded that the managers and leaders of firms need to develop their business models in such a way that it supports the triple bottom line performance of the firm so that they can achieve sustainability. This study would help managers of firms to realize that they should reexamine their leadership practices and behaviors with the consideration of the triple bottom line framework and integrate the sustainable approach and CSR practice into their business strategy so that they can accomplish the goal of sustainability for business and fulfill the social demands and expectations. (shrink)

The increasing use of opioid treatment agreements has prompted debate within the medical community about ethical challenges with respect to their implementation. The focus of debate is usually on the efficacy of OTAs at reducing opioid misuse, how OTAs may undermine trust between physicians and patients and the potential coercive nature of requiring patients to sign such agreements as a condition for receiving pain care. An important consideration missing from these conversations is the potential for racial bias in the current (...) way that OTAs are incorporated into clinical practice and in the amount of physician discretion that current opioid guidelines support. While the use of OTAs has become mandatory in some states for certain classes of patients, physicians are still afforded great leeway in how these OTAs are implemented in clinical practice and how their terms should be enforced. This paper uses the guidelines provided for OTA implementation by the states of Indiana and Pennsylvania as case studies in order to argue that giving physicians certain kinds of discretion may exacerbate racial health disparities. This problem cannot simply be addressed by minimising physician discretion in general, but rather by providing mechanisms to hold physicians accountable for how they treat patients on long-term opioid therapy to ensure that such treatment is equitable. There are no data in this work. (shrink)

We consider the first-order finite-difference expression of the commutator between d / dx and x. This is the appropriate setting in which to propose commutators and time operators for a quantum system with an arbitrary potential function and a discrete energy spectrum. The resulting commutators are identified as universal lowering and raising operators. We also find time operators which are finite-difference derivations with respect to the energy. The matrix elements of the commutator in the energy representation are analyzed, (...) and we find consistency with the equality \. We apply the theory to the particle in an infinite well and for the Harmonic oscillator as examples. (shrink)

The first-order temporal logics with □ and ○ of time structures isomorphic to ω (discrete linear time) and trees of ω-segments (linear time with branching gaps) and some of its fragments are compared: the first is not recursively axiomatizable. For the second, a cut-free complete sequent calculus is given, and from this, a resolution system is derived by the method of Maslov.

This article deals with the ordering of interim measures in the arbitration process. Under the Lithuanian law, only one interim measure can be applied by commercial arbitration courts. It involves granting an order to pay the established amount as a deposit. This measure cannot be enforced unless the party acts in good faith and agrees to pay the deposit. However, if a party ignores the decision of the arbitration court, a third party court may apply to the state court (...) with a request to order interim measures provided for in the Lithuanian Code of Civil Procedure. Under the existing Lithuanian legislation, state courts have the exclusive right to order enforceable interim measures. The present situation is not satisfactory and should be changed. The legislator is now considering a new draft law on the Lithuanian commercial arbitration. The draft law provides for changes in ordering interim measures. The proposed regulation is in compliance with the goals of Article 17 of UNICITRAL Model Law on Commercial Arbitration. However, compliance with the rules of UNCITRAL is not the main goal of the proposed regulation. With the new regulation, the legislator aims for more efficient commercial arbitration with more extensive scope that will allow the commercial arbitration court to order interim measures independently. By allowing the commercial arbitration court to order interim measures independently, the legislator extends credibility of arbitration and makes it more inviting for the parties. Under the draft law on the Lithuanian commercial arbitration, state courts and commercial arbitration courts as discretionary institutions shall be entitled to order interim measures. On the other hand, a decision or a ruling of the commercial arbitration court shall be enforced if an execution order is issued by the state court. In this case, the state court shall be entitled to review whether the content of the decision or ruling of the commercial arbitration court complies with the imperative rules of law or public order. According to the proposed draft law on the Lithuanian commercial arbitration, both state and commercial arbitration courts shall have an independent right of discretion to order interim measures. However, some differences exist. In cases heard by commercial arbitration courts interim measures shall be ordered, whereas state courts acting under applications for interim measures filed by the parties or commercial arbitration courts, irrespective of the location of the arbitration process, shall be entitled to order interim measures. (shrink)

The observed complexity of nature is often attributed to an intrinsic propensity of matter to self-organize under certain (e.g., dissipative) conditions. In order better to understand and test this vague thesis, we define complexity as “logical depth,” a notion based on algorithmic information and computational time complexity. Informally, logical depth is the number of steps in the deductive or causal path connecting a thing with its plausible origin. We then assess the effects of dissipation, noise, and spatial and other (...) symmetries of the initial conditions and equations of motion on the asymptotic complexity-generating abilities of statistical-mechanical model systems. We concentrate on discrete, spatially-homogeneous, locally-interacting systems such as kinetic Ising models and cellular automata. (shrink)

It is shown that the infinite-valued first-order Gödel logic G° based on the set of truth values {1/k: k ε w {0}} U {0} is not r.e. The logic G° is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kröger's temporal logic of programs (even of the fragment without the nexttime operator O) and of (...) the authors' temporal logic of linear discrete time with gaps follows. (shrink)

According to Peirce one of the most important philosophical problems is continuity. Consequently, he set forth an innovative and peculiar approach in order to elucidate at once its mathematical and metaphysical challenges through proper non-classical logical reasoning. I will restrain my argument to the definition of the different types of discrete collections according to Peirce, with a special regard to the phenomenon called ?premonition of continuity? (Peirce, 1976, Vol. 3, p. 87, c. 1897).

Temporal stability and change in neutral contexts can transform continuous experiences into distinct and memorable events. However, less is known about how shifting emotional states influence these memory processes, despite ample evidence that emotion impacts non-temporal aspects of memory. Here, we examined if emotional stimuli influence temporal memory for recent event sequences. Participants encoded lists of neutral images while listening to auditory tones. At regular intervals within each list, participants heard emotional positive, negative, or neutral sounds, which served as “emotional (...) event boundaries” that divided each sequence into discrete events. Temporal order memory was tested for neutral item pairs that either spanned an emotional sound or were encountered within the same auditory event. Encountering a highly arousing event boundary led to faster response times for items encoded within the next event. Critically, we found that highly arousing sounds had different effects on binding ongoing versus ensuing sequential representations in memory. Specifically, highly arousing sounds were significantly more likely to enhance temporal order memory for ensuing information compared to information that spanned those boundaries, especially for boundaries with negative valence. These findings suggest that within aversive emotional contexts, fluctuations in arousal help shape the temporal organisation of events in memory. (shrink)

The formal properties of orbits in a plane are explored by elementary topology. The notions developed from first principles include: convex and polygonal orbits; convexity; orientation, winding number and interior; convex and star-shaped regions. It is shown that an orbit that is convex with respect to each of its interior points bounds a convex region. Also, an orbit that is convex with respect to a fixed point bounds a star-shaped region.Biological considerations that directed interest to these patterns are indicated, and (...) the implications of the prospect of higher orders of star-shapedness mentioned. (shrink)

This short paper is meant to be an introduction to the ‘Letter to Alan Turing’ that follows it. It summarizes some basic ideas in information theory and very informally hints at their mathematical properties. In order to introduce Turing’s two main theoretical contributions, in Theory of Computation and in Morphogenesis, the fundamental divide between discrete vs. continuous structures in mathematics is presented, as it is also a divide in his scientific life. The reader who is familiar with these (...) notions, and is convinced that they are relevant in the mathematical understanding of phenomena, may skip this introduction and go directly to the Letter. (shrink)

This paper is concerned with the event-based state and fault estimation problem for a class of linear discrete systems with randomly occurring faults and missing measurements. Different from the static event-based transmission mechanism with a constant threshold, a dynamic event-based mechanism is exploited here to regulate the threshold parameter, thus further reducing the amount of data transmission. Some mutually independent Bernoulli random variables are used to characterize the phenomena of ROFs and missing measurements. In order to simultaneously estimate (...) the system state and the fault signals, the main attention of this paper is paid to the design of recursive filter; for example, for all DETM, ROFs, and missing measurements, an upper bound for the estimation error covariance is ensured and the relevant filter gain matrix is designed by minimizing the obtained upper bound. Moreover, the rigorous mathematical analysis is carried out for the exponential boundedness of the estimation error. It is clear that the developed algorithms are dependent on the threshold parameters and the upper bound together with the probabilities of missing measurements and ROFs. Finally, a numerical example is provided to indicate the effectiveness of the presented estimation schemes. (shrink)

We deal with the design problem of nonfragile state estimator for discrete-time genetic regulatory networks with time-varying delays and randomly occurring uncertainties. In particular, the norm-bounded uncertainties enter into the GRNs in random ways in order to reflect the characteristic of the modelling errors, and the so-called randomly occurring uncertainties are characterized by certain mutually independent random variables obeying the Bernoulli distribution. The focus of the paper is on developing a new nonfragile state estimation method to estimate the (...) concentrations of the mRNA and the protein for considered uncertain delayed GRNs, where the randomly occurring estimator gain perturbations are allowed. By constructing a Lyapunov-Krasovskii functional, a delay-dependent criterion is obtained in terms of linear matrix inequalities by properly using the discrete-time Wirtinger-based inequality and reciprocally convex combination approach as well as the free-weighting matrix method. It is shown that the proposed method ensures that the estimation error dynamics is globally asymptotically stable and the desired estimator parameter is designed via the solutions to certain LMIs. Finally, we provide two numerical examples to illustrate the feasibility and validity of the proposed estimation results. (shrink)

ABSTRACT In this paper we present a new semantic approach for propositional linear temporal logic with discrete time, strongly based in the well-order of IN (the set of natural numbers). We consider temporal connectives which express precedence, posteriority and simultaneity, and they provide a family of expressively complete temporal logics. The selection of the new semantics and connectives used in this work was principally to obtain a suitable executable temporal logic, which can be used for the specification and (...) control of process behaviour in discrete time in a similar way to thai presented by D. Gabbay in [GAB 89], Our new approach has two advantages: firstly, the connectives are defined intuitively so that they have interpretations which relate to properties of interest in real systems; secondly, it provides a new semantics that facilitates simpler proofs of many valid formulas and metatheorems. To confirm this second advantage, we use our semantics to give a formal proof of the Separation Theorem for one of these logics (specifically LN3). The great interest of this Separation Theorem for the executable temporal logic is emphasised by D. Gabbay in [GAB 89], [GAB 88]. (shrink)

A totally ordered group G is essentially periodic if for every definable non-trivial convex subgroup H of G every definable subset of G is equal to a finite union of cosets of subgroups of G on some interval containing an end segment of H; it is coset-minimal if all definable subsets are equal to a finite union of cosets, intersected with intervals. We study definable sets and functions in such groups, and relate them to the quasi-o-minimal groups introduced in Belegradek (...) et al. . Main results: An essentially periodic group G is abelian; if G is discrete, then definable functions in one variable are ultimately piecewise linear. A group such that every model elementarily equivalent to it is coset-minimal is quasi-o-minimal , and its definable functions in one variable are piecewise linear. (shrink)

In this paper we study the logical strength of the determinacy of infinite binary games in terms of second order arithmetic. We define new determinacy schemata inspired by the Wadge classes of Polish spaces and show the following equivalences over the system RCA0*, which consists of the axioms of discrete ordered semi‐rings with exponentiation, Δ10 comprehension and Π00 induction, and which is known as a weaker system than the popularbase theory RCA0: 1. Bisep(Δ10, Σ10)‐Det* ↔ WKL0, 2. Bisep(Δ10, (...) Σ20)‐Det* ↔ ATR0 + Σ11 induction, 3. Bisep(Σ10, Σ20)‐Det* ↔ Sep(Σ10, Σ20)‐Det* ↔ Π11‐CA0, 4. Bisep(Δ20, Σ20)‐Det* ↔ Π11‐TR0, where Det* stands for the determinacy of infinite games in the Cantor space (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

This paper poses a puzzle for contemporary Kantian political philosophy. Kantian political philosophers hold that the state’s purpose is to secure the conditions for people’s innate right to equal freedom, while at the same time claiming that innate right does not give a determinate set of conditions that the state is to bring about. Officials, then, have to make decisions in cases where the considerations of innate right provide no further guidance. I argue that, intuitively, in such cases there are (...) (i) some further considerations that officials may appeal to and (ii) some further considerations that officials may not appeal to in order to decide among the options consistent with people’s innate right and then raise difficulties for the ability of current Kantian accounts to explain how they can accept both (i) and (ii). I conclude by suggesting one potential path forward for Kantians to address this puzzle. (shrink)

This paper questions the dogmatic stance of the domestic courts toward mandatory orders for treatment, arguing that this has the potential to subjugate patients' interests to clinical discretion, and proposing a via media to accommodate the legitimate concerns of all parties.

In constructive mathematics it is of interest to consider a more general, but classically equivalent, notion of linear order, a so-called pseudo-order. The prime example is the order of the constructive real numbers. We examine two kinds of constructive completions of pseudo-orders: order completions of pseudo-orders and Cauchy completions of ordered groups and fields. It is shown how these can be predicatively defined in type theory, also when the underlying set is non-discrete. Provable choice principles, (...) in particular a generalisation of dependent choice, are used for showing set-representability of cuts. (shrink)

This paper investigates the event-based consensus problem for the heterogeneous hybrid multiagent system. First, the heterogeneous hybrid MAS is proposed which contains continuous and discrete-time subsystems with second-order and first-order heterogeneous dynamics. Second, the event-triggered protocols are proposed, which mainly include the event-based control laws and event-triggered conditions for different kinds of agents. Then, the consensus conclusions of fixed topology and switching topologies are obtained based on graph theory and nonnegative matrix theory, which include the constraints on (...) control parameters, coupling gains, and sampling interval to guarantee consensus. Finally, a simulation example is given to verify the efficiency of the proposed protocols. (shrink)

Scientific observation has led to the discovery of recurring patterns in nature. Symmetry is the property of an object showing regularity in parts on a plane or around an axis. There are several types of symmetries observed in the natural world and the most common are mirror symmetry, radial symmetry, and translational symmetry. Symmetries can be continuous or discrete. A discrete symmetry is a symmetry that describes non-continuous changes in an object. A continuous symmetry is a repetition of (...) an object an infinite number of times. A consequence of continuous symmetry is the existence of conservation laws. A natural system with discrete and continuous symmetries displays several physical properties, such as the existence of long-range order. Geometric shapes exhibit symmetry as mirror reflections of each other. Symmetry in nature has been a model of beauty since the beginning of civilization. Since the earliest times, nature itself has manifestly been a model, evincing regularity in sundry forms and occurrences–from minerals and plants to the anatomy of living beings, to the regularly recurring stellar constellations. For Ancient Greek philosophers, proportion, symmetry, and harmony, were the basics to determine whether something is beautiful or not. Ancient Greek philosophers were known for bringing logic and rational thinking to phenomena that were previously explained by mythology and Gods. They observed the natural world around them and used their knowledge to answer questions about the proportion in observable objects and the origin of Earth. The Greek words _summetria_ and _summetros_ appear frequently in the Timaeus, defined as parts with each other and with the whole. Plato has a very rough concept of symmetry, and when he uses “beauty” to characterize the so-called Platonic Solids in the Timaeus, he seems to be emphasizing their regularity and indirectly their symmetry. Plato believes the four elements (earth, air, fire, water) have been constructed by the Demiurge, or a divine craftsman who appoints order in an otherwise chaotic universe. Minerals are representative examples of beauty, order, and symmetry in inorganic materials. Well-formed minerals (crystals) are a collection of equivalent faces related by symmetry. The goal of this paper is to relate the perspective of symmetry in ancient Greek philosophy with the modern scientific evidence of the geometric description of crystal structures. The five Platonic solids are ideal models of geometrical patterns that occur throughout the world of minerals. In this paper, minerals serve as a model of connecting the symmetry theory in Plato’s philosophy and modern advances in mineralogy. (shrink)

This paper brings together a most fascinating and under-examined body of early New World writing that belong to a genre of writing I call “the torture narrative” with the insights of Marshall McLuhan in order to offer a way of thinking about body parts, especially hands, teeth, tongues, and eyeballs, and their extensions through technologies such as alphabets, manuscripts, books, and weapons. At its core are questions about the nature and effects of the changes wrought by the early-Gutenberg era—a (...) period characterized by vast scientific and technical discoveries, rising nationalisms, explorations, and conquests—in the New World. Rather than explore the ways an early New World print culture reflects or distorts discrete historical or ethnographic facts about colonial contact, I want to probe the ways the torture narrative speaks to the conditions of speaking and writing, and therefore to the discursive and textual production of the New World subject and New World space. My inquiry seeks to understand both the violence in, and the violence of, representation at founding historical moments, but its relevance far exceeds Renaissance or American studies. At stake in this discussion is an appreciation of a mode of intellectual inquiry that is attentive to the relationship between media technologies and the social worlds they reflect and transform. (shrink)

The study of the chaotic dynamics in fractional-order discrete-time systems has received great attention over the last years. Some efforts have been also devoted to analyze fractional maps with special features. This paper makes a contribution to the topic by introducing a new fractional map that is characterized by both particular dynamic behaviors and specific properties related to the system equilibria. In particular, the conceived one dimensional map is algebraically simpler than all the proposed fractional maps in the (...) literature. Using numerical simulation, we investigate the dynamic and complexity of the fractional map. The results indicate that the new one-dimensional fractional map displays various types of coexisting attractors. The approximate entropy is used to observe the changes in the sequence sequence complexity when the fractional order and system parameter. Finally, the fractional map is applied to the problem of encrypting electrophysiological signals. For the encryption process, random numbers were generated using the values of the fractional map. Some statistical tests are given to show the performance of the encryption. (shrink)