Barycentres and metric properties

We present some metric problems, where central tools are barycentres and Leibniz's scalar function. Two of the examples are not directly related to geometry, showing the generality of this way of mathematical thinking.     

Leibniz's rule with n = ‐1

This paper derives Leibniz's rule with n = ‐1 and shows that it may be used to simplify extended integration by parts in certain examples which occur in the derivation of Fourier series. A simple derivation of the general Taylor expansion with integral remainder form is also obtained.     

Coincident Entities and Question-Begging Predicates: an Issue in Meta-Ontology

Meta-ontology (in van Inwagen's sense) concerns the methodology of ontology, and a controversial meta-ontological issue is to what extent ontology can rely on linguistic analysis while establishing the furniture of the world. This paper discusses an argument advanced by some ontologists (I call them unifiers) against supporters of or coincident entities (I call them multipliers) and its meta-ontological import. Multipliers resort to Leibniz's Law to establish that spatiotemporally coincident entities a and b are distinct, by pointing at a predicate F() made true by a and false by b. Unifiers try to put multipliers in front of a dilemma: in attempting to introduce metaphysical differences on the basis of semantic distinctions, multipliers either (a) rest on a fallacy of verbalism, entailed by a trade-off between a de dicto and a de re reading of modal claims, or (b) beg the question against unifiers by having to assume the distinction between a and b beforehand. I shall rise a tu quoque, showing that unifiers couldn't even distinguish material objects (or events) from the spatiotemporal regions they occupy unless they also resorted to linguistic distinctions. Their methodological aim to emancipate themselves from linguistic analysis in ontological businesses is therefore problematic.     

Beyond Cartesian limits: Leibniz's passage from algebraic to “transcendental” mathematics

This article deals with Leibniz's reception of Descartes' “geometry.” Leibnizian mathematics was based on five fundamental notions: calculus, characteristic, art of invention, method, and freedom. On the basis of methodological considerations Leibniz criticized Descartes' restriction of geometry to objects that could be given in terms of algebraic (i.e., finite) equations: “Descartes's mind was the limit of science.” The failure of algebra to solve equations of higher degree led Leibniz to develop linear algebra, and the failure of algebra to deal with transcendental problems led him to conceive of a science of the infinite. Hence Leibniz reconstructed the mathematical corpus, created new (transcendental) notions, and redefined known notions (equality, exactness, construction), thus establishing “a veritable complement of algebra for the transcendentals”: infinite equations, i.e., infinite series, became inestimable tools of mathematical research.     

A generalization of Mühlbach's extension principle for determinantal identities

Mühlbach's extension principle for determinantal identities generalizes Muir's law of extensible minors. Some particular issues with Mühlbach–Beckermann's identity [A general determinantal identity of Sylvester type and some applications, Linear Algebra Appl. 197, 198 (1994), pp. 93–112] led to the conjecture of a more general extension method than Mühlbach's. However, no confirmation seems to have been reported so far. In this note, we present a generalization of Mühlbach's extension principle which confirms that conjecture. The whole identity of Mühlbach–Beckermann is put in a simpler form from which a new interpretation as an extension of Leibniz's definition of a determinant.     

Interpolation and Combinatorial Functions

Using some basic results about polynomial interpolation, divided differences, and Newton polynomial sequences we develop a theory of generalized binomial coefficients that permits the unified study of the usual binomial coefficients, the Stirling numbers of the second kind, the q-Gaussian coefficients, and other combinatorial functions. We obtain a large number of combinatorial identities as special cases of general formulas. For example, Leibniz's rule for divided differences becomes a Chu-Vandermonde convolution formula for each particular family of generalized binomial coefficients.     

Class numbers of quadratic function fields and continued fractions

A function field version of a theorem of F. Hirzebruch relating continued fractions to class numbers of quadratic number fields is established. Our approach is based on Artin's thesis and Zagier's proof of Hirzebruch's theorem. Some of our results seem to be of independent interest, e.g. explicit formulas for Zeta functions of real quadratic function fields.     

Is the statement of Murphy's law valid?

Murphy's Law is not a law in the formal sense yet popular science often compares it with the Second Law of Thermodynamics as both the statements point toward a more disorganized state with time. In this article, we first construct a mathematically equivalent statement for Murphy's Law and then disprove it using the intuitive idea that energy differences will level off along the paths of steepest descent, or along trajectories of least action. © 2015 Wiley Periodicals, Inc. Complexity 21: 374–380, 2016     

The chaotic universe

In this paper we suggest a formulation that would bear out the spirit of Prigogine's “Order Out of Chaos” and Wheeler's “Law Without Law”. In it a typical elementary particle length, namely the pion Compton wavelength arises from the random motion of the N particles in the universe of dimension R. It is then argued in the light of recent work that this is the origin of the laws of physics and leads to a cosmology consistent with observation.     

Some weak and strong laws of large numbers for D[0,1]-valued random variables

Pointwise Weak Law of Large Numbers and Weak Law of Large Numbers in the norm topology of D[0,l] are shown to be equivalent under uniform convex tightness and uniform integrability conditions for weighted sums of a sequence of random elements in D[0,1]. Uniform convex tightness and uniform integrability conditions are jointly characterized. Marcinkiewicz–Zygmund–Kolmogorov's and Brunk– Chung's Strong Laws of Large Numbers are derived in the setting of D[0,l]-space under uniform convex tightness and uniform integrability conditions. Equivalence of pointwise convergence, convergence in the Skorokhod topology and convergence in the norm topology f o r sequences in D[0,l] is studied     

Quasigroups and tactical systems

A quasigroupQ is a set together with a binary operation which satisfies the condition that any two elements of the equationxy =z uniquely determines the third. A quasigroup is in indempotent when any elementx satisfies the indentityxx =x. Several types of Tactical Systems are defined as arrangement of points into “blocks” in such a way as to balance the incidence of (ordered or unordered) pairs of points, and shown to be coexistent with idempotent quasigroups satisfying certain identifies. In particular the correspondences given are: 1. totally symmetric idempotent quasigroups and Steiner triple systems, 2. semi-symmetric idempotent quasigroups and directed triple systems, 3. idempotent quasigroups satisfying Schröder's Second Law, namely (xy)(yx)=x, and triple tourna-ments, and 4. idempotent quasigroups satisfying Stein's Third Law, namely (xy)(yx)=y, and directed tournaments. These correspondences are used to obtain corollaries on the existence of such quasig-roups from constructions of the Tactical Systems. In particular this provides a counterexample to an ”almost conjecture“ of Norton and Stein (1956) concerning the existence of those quasigroups in 3 and 4 above. Indeed no idempotent qnasigroups satisfying Stein's Third Law and with order divisible by four were known to N. S. Mendelsohn when he wrote a paper on such quasigroups for the Third Waterloo Conference on Combinatorics (May, 1968). Finally, a construction for triple tournaments is interpreted as a Generalized Semi-Direct Product of idempotent quasigroups.     

The geometric origin of perspectivist science in G.W. Leibniz. Analysis based on unpublished manuscripts

The perspectivist research carried out by G.W. Leibniz between 1679 and 1686 in the field of geometry is analysed. This work is reflected in six, as yet unpublished, texts, of which the main three are analysed: Constructio et usus scalae perspectivae, Origo regularum artis perspectivae and Scientia perspectiva. The philosophical perspectivism advocated by the German thinker is widely known, but his geometric research on perspective is much less so. This article seeks to remedy this situation. The first of these writings (Constructio et usus scalae perspectivae) includes Leibniz's experimentation with the perspectivist methodology of scales. Then, in Origo regularum artis perspectivae quales, Leibniz constructs his perspectivist regula generalis. Finally, Leibniz wrote Scientia perspectiva and readdresses the main rule of perspective and experiments with the theoretical limits of the analysis carried out in this discipline. Primarily, he supposed a ‘minimum distance’ between the elements that constitute it, and then theorised an ‘infinite interval’ between these same elements.     

The Integration of Math and Science Via Centroids

Science problems enhance and promote math functions to establish some formulas to solve them; conversely, many math results give the explanation and the development of science phenomena and their related situation. From Archimedes’ Law of the Lever, together with some properties of vector's representation, the geometrical construction of the weighted centroid of gravity of finite particles is given, the new proving of Ceva's and Menelaus's results is explored, and a related result to spacial shape is set up. These presentations are important in math, physics, chemistry, statistics, and engineering. The ideas are significant to these fields for the integration of multifarious curricula.     

Strassen's local law of the iterated logarithm for Lévy's area

We prove a Strassen's law of the iterated logarithm at zero for Lévy's area process. Contrary to the Brownian case, the time inversion argument doesn't seem to work. Here, the main tool in the proof is large deviations estimates for diffusion processes with small diffusion coefficients.     

Newton's second law in field theory

In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called the geodesic k-vector field, analogous to the ordinary geodesic field and which describes the inertial motions (i.e., evolution in the absence of forces). From this generalized Newton's law, the corresponding Hamilton's canonical equations of field theory (Hamilton-De Donder-Weyl equations) are obtained by a simple procedure. It is shown that solutions of generalized Newton's equation also hold the canonical equations. However, unlike the ordinary case, Newton equations determined by different forces can define equal Hamilton's equations.     

A distributional form of Little's Law

For many system contexts for which Little's Law is valid a distributional form of the law is also valid. This paper establishes the prevalence of such system contexts and makes clear the value of the distributional form.     

On Drinfeld's DG quotient

The possibility of constructing quotients of differential graded (= dg) categories is essential in non-commutative algebraic geometry. The first construction of dg quotients appeared in Keller's work (Keller (1994) [21]) and it was recently followed by Drinfeld's elegant approach (Drinfeld (2004) [9]). Although Drinfeld's dg quotient admits a very simple construction, it didn't seem to be intrinsically defined. In this article we complete this aspect of Drinfeld's work by providing three different characterizations of Drinfeld's dg quotient in terms of simple universal properties.     

Branching Brownian Motion with Catalytic Branching at the Origin

We consider a branching Brownian motion in which binary fission takes place only when particles are at the origin at a rate β>0 on the local time scale. We obtain results regarding the asymptotic behaviour of the number of particles above λt at time t, for λ>0. As a corollary, we establish the almost sure asymptotic speed of the rightmost particle. We also prove a Strong Law of Large Numbers for this catalytic branching Brownian motion.     

The Non-Commutative Cycle Lemma

We present a non-commutative version of the Cycle Lemma of Dvoretsky and Motzkin that applies to free groups and use this result to solve a number of problems involving cyclic reduction in the free group. We also describe an application to random matrices, in particular the fluctuations of Kesten's Law.     

Student interpretations of the terms in first-order ordinary differential equations in modelling contexts

A study of first-year undergraduate students′ interpretational difficulties with first-order ordinary differential equations (ODEs) in modelling contexts was conducted using a diagnostic quiz, exam questions and follow-up interviews. These investigations indicate that when thinking about such ODEs, many students muddle thinking about the function that gives the quantity to be determined and the equation for the quantity's rate of change, and at least some seem unaware of the need for unit consistency in the terms of an ODE. It appears that shifting from amount-type thinking to rates-of-change-type thinking is difficult for many students. Suggestions for pedagogical change based on our results are made.