scholarly journals Cartesian Difference Categories

2020 ◽  
pp. 57-76
Author(s):  
Mario Alvarez-Picallo ◽  
Jean-Simon Pacaud Lemay
Keyword(s):  
Finite Differences ◽  
Tangent Bundle ◽  
Differential Calculus ◽  
Directional Derivative ◽  
The Other ◽  
Smooth Functions ◽  
Kleisli Category ◽  
Categorical Models ◽  
Action Models ◽  
Differential Categories

AbstractCartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential $$\lambda $$ λ -calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation such as the calculus of finite differences or the Boolean differential calculus. On the other hand, change action models have been shown to capture these examples as well as more “exotic” examples of differentiation. However, change action models are very general and do not share the nice properties of a Cartesian differential category. In this paper, we introduce Cartesian difference categories as a bridge between Cartesian differential categories and change action models. We show that every Cartesian differential category is a Cartesian difference category, and how certain well-behaved change action models are Cartesian difference categories. In particular, Cartesian difference categories model both the differential calculus of smooth functions and the calculus of finite differences. Furthermore, every Cartesian difference category comes equipped with a tangent bundle monad whose Kleisli category is again a Cartesian difference category.

scholarly journals Cartesian Difference Categories

2021 ◽  
Vol Volume 17, Issue 3 ◽  
Author(s):  
Mario Alvarez-Picallo ◽  
Jean-Simon Pacaud Lemay
Keyword(s):  
Finite Differences ◽  
Differential Calculus ◽  
Directional Derivative ◽  
Lambda Calculus ◽  
Smooth Functions ◽  
Kleisli Category ◽  
Discrete Nature ◽  
Categorical Models ◽  
Action Models ◽  
Differential Categories

Cartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential $\lambda$-calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation of a more discrete nature such as the calculus of finite differences. On the other hand, change action models have been shown to capture these examples as well as more "exotic" examples of differentiation. But change action models are very general and do not share the nice properties of Cartesian differential categories. In this paper, we introduce Cartesian difference categories as a bridge between Cartesian differential categories and change action models. We show that every Cartesian differential category is a Cartesian difference category, and how certain well-behaved change action models are Cartesian difference categories. In particular, Cartesian difference categories model both the differential calculus of smooth functions and the calculus of finite differences. Furthermore, every Cartesian difference category comes equipped with a tangent bundle monad whose Kleisli category is again a Cartesian difference category.

scholarly journals Exponential Functions in Cartesian Differential Categories

2020 ◽  
Author(s):  
Jean-Simon Pacaud Lemay
Keyword(s):  
Differential Equations ◽  
Exponential Function ◽  
Differential Calculus ◽  
Exponential Map ◽  
Dual Numbers ◽  
Exponential Functions ◽  
Smooth Functions ◽  
Exponential Maps ◽  
Differential Categories ◽  
Complex Exponential

Abstract In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function $$e^x$$ e x from classical differential calculus. A differential exponential map is an endomorphism which is compatible with the differential combinator in such a way that generalizations of $$e^0 = 1$$ e 0 = 1 , $$e^{x+y} = e^x e^y$$ e x + y = e x e y , and $$\frac{\partial e^x}{\partial x} = e^x$$ ∂ e x ∂ x = e x all hold. Every differential exponential map induces a commutative rig, which we call a differential exponential rig, and conversely, every differential exponential rig induces a differential exponential map. In particular, differential exponential maps can be defined without the need of limits, converging power series, or unique solutions of certain differential equations—which most Cartesian differential categories do not necessarily have. That said, we do explain how every differential exponential map does provide solutions to certain differential equations, and conversely how in the presence of unique solutions, one can derivative a differential exponential map. Examples of differential exponential maps in the Cartesian differential category of real smooth functions include the exponential function, the complex exponential function, the split complex exponential function, and the dual numbers exponential function. As another source of interesting examples, we also study differential exponential maps in the coKleisli category of a differential category.

scholarly journals Teacher knowledge of error analysis in differential calculus

Pythagoras ◽  
2014 ◽  
Vol 35 (2) ◽  
Author(s):  
Eunice K. Moru ◽  
Makomosela Qhobela ◽  
Poka Wetsi ◽  
John Nchejane
Keyword(s):  
Mathematics Education ◽  
Data Collection ◽  
Error Analysis ◽  
Teacher Knowledge ◽  
Conceptual Understanding ◽  
Differential Calculus ◽  
The Other ◽  
Multiple Perspectives ◽  
Error Identification ◽  
Knowledge For Teaching

The study investigated teacher knowledge of error analysis in differential calculus. Two teachers were the sample of the study: one a subject specialist and the other a mathematics education specialist. Questionnaires and interviews were used for data collection. The findings of the study reflect that the teachers’ knowledge of error analysis was characterised by the following assertions, which are backed up with some evidence: (1) teachers identified the errors correctly, (2) the generalised error identification resulted in opaque analysis, (3) some of the identified errors were not interpreted from multiple perspectives, (4) teachers’ evaluation of errors was either local or global and (5) in remedying errors accuracy and efficiency were emphasised more than conceptual understanding. The implications of the findings of the study for teaching include engaging in error analysis continuously as this is one way of improving knowledge for teaching.

scholarly journals Bistructures, Bidomains and Linear Logic

1994 ◽  
Vol 1 (9) ◽  
Author(s):  
Gordon Plotkin ◽  
Glynn Winskel
Keyword(s):  
Partial Order ◽  
Full Subcategory ◽  
Linear Logic ◽  
The Other ◽  
Causal Dependency ◽  
Input And Output ◽  
Event Structures ◽  
Kleisli Category ◽  
Higher Types ◽  
Cartesian Closed

Bistructures are a generalisation of event structures to represent spaces of functions at higher types; the partial order of causal dependency is replaced by two orders, one associated with input and the other output in the behaviour of functions. Bistructures form a categorical model of Girard's classical linear logic in which the involution of linear logic is modelled, roughly speaking, by a reversal of the roles of input and output. The comonad of the model has associated co-Kleisli category which is equivalent to a cartesian-closed full subcategory of Berry's bidomains.

scholarly journals A note on the turning point for the quadratic trend

2019 ◽  
Vol 5 (2) ◽  
pp. 39-48
Author(s):  
Vedran Kojić ◽  
Tihana Škrinjarić
Keyword(s):  
Statistical Model ◽  
Unemployment Rate ◽  
Differential Calculus ◽  
Turning Point ◽  
Quadratic Function ◽  
The Other ◽  
New Method ◽  
Quadratic Trend ◽  
Function Finding

AbstractThe quadratic trend is a statistical model described by the quadratic function. Finding its extremum (also called the vertex or the turning point) using differential calculus or completing the square method is very well known in the literature. In this paper, a new method for finding the extremum of the quadratic function, based on a simple mathematical inequality is proposed. In comparison with the other two known methods, our method does not require the differentiability assumption and it takes fewer steps than completing the square method. Also, it is shown how the turning point for the quadratic trend can be applied in forecasting the unemployment rate in Croatia in the first quarter of 2019. The obtained conclusions are equal to the conclusions obtained in the usual way by using forecasting software.

Coupon Bond Duration and Convexity Analysis

2021 ◽  
pp. 316-345
Author(s):  
Vedran Kojić ◽  
Margareta Gardijan Kedžo ◽  
Zrinka Lukač
Keyword(s):  
Mathematical Analysis ◽  
Empirical Data ◽  
Differential Calculus ◽  
Risk Measures ◽  
The Other ◽  
Complete Analysis ◽  
Elementary Algebra ◽  
Convexity Properties ◽  
Convexity Analysis ◽  
Coupon Bond

Coupon bond duration and convexity are the primary risk measures for bonds. Given their importance, there is abundant literature covering their analysis, with calculus being used as the dominant approach. On the other hand, some authors have treated coupon bond duration and convexity without the use of differential calculus. However, none of them provided a complete analysis of bond duration and convexity properties. Therefore, this chapter fills in the gap. Since the application of calculus may be complicated or even inappropriate if the functions in question are not differentiable (as indeed is the case with the bond duration and convexity functions), in this chapter the properties of bond duration and convexity functions by using elementary algebra only are proved. This provides an easier way of approaching this problem, thus making it accessible to a wider audience not necessarily familiar with tools of mathematical analysis. Finally, the properties of these functions are illustrated by using empirical data on coupon bonds.

scholarly journals Dilation volumes of sets of finite perimeter

2018 ◽  
Vol 50 (4) ◽  
pp. 1095-1118
Author(s):  
Markus Kiderlen ◽  
Jan Rataj
Keyword(s):  
Directional Derivative ◽  
Dimensional Euclidean Space ◽  
Functions Of Bounded Variation ◽  
Smooth Functions ◽  
Area Measure ◽  
Structuring Element ◽  
Finite Perimeter ◽  
Contact Distribution ◽  
Contact Distribution Function ◽  
The Right

Abstract In this paper we analyze the first-order behavior (that is, the right-sided derivative) of the volume of the dilation A⊕tQ as t converges to 0. Here A and Q are subsets of n-dimensional Euclidean space, A has finite perimeter, and Q is finite. If Q consists of two points only, n and n+u, say, this derivative coincides up to a sign with the directional derivative of the covariogram of A in direction u. By known results for the covariogram, this derivative can therefore be expressed by the cosine transform of the surface area measure of A. We extend this result to finite sets Q and use it to determine the derivative of the contact distribution function with finite structuring element of a stationary random set at 0. The proofs are based on an approximation of the indicator function of A by smooth functions of bounded variation.

VII—Tidal Charts Based on Coastal Data: Irish Sea

1953 ◽  
Vol 64 (1) ◽  
pp. 90-111
Author(s):  
A. T. Doodson ◽  
J. R. Rossiter ◽  
R. H. Corkan
Keyword(s):  
Shallow Water ◽  
Finite Differences ◽  
The Other ◽  
Irish Sea ◽  
Successive Approximations ◽  
Relaxation Methods ◽  
The World ◽  
Frictional Forces ◽  
Two Phases ◽  
Tidal Streams

SynopsisThe object of this paper is not to produce a chart of co-tidal and co-range lines which is more accurate than an existing one, but to investigate methods of computing such charts on the supposition that there are no observations of tidal streams such as were used to produce the existing chart. Only coastal observations of tidal elevations are supposed to be known, for such conditions would exist in many parts of the world. The methods used are similar to the so-called “relaxation methods”, using finite differences in all variables and attempting to satisfy all the conditions of motion within the sea, proceeding by successive approximations. There are many difficulties, peculiar to the tidal problem, in the application of these methods, due to the very irregular coast-lines and depths, gaps in the coasts, shallow water near the coasts, frictional forces, and the very serious complication due to the fact that the tides are oscillating and thus require two phases to be investigated simultaneously owing to their reactions one upon the other. One very important point in testing the methods is that no use whatever should be made of existing charts in obtaining first approximations of heights to commence the processes, not even where there are wide entrances to the sea. The resulting chart is shown to be very closely the same as the existing chart, thus proving the validity of the method.

scholarly journals XVIII. On the formulæ investigated by Dr. Brinkley for the general term in the deve­lopment of Lagrange’s expression for the summation of series and for successive integrations

1860 ◽  
Vol 150 ◽  
pp. 319-323
Keyword(s):  
General Expression ◽  
Finite Differences ◽  
Royal Society ◽  
General Term ◽  
Numerical Results ◽  
The Other ◽  
Exponential Functions ◽  
Summation Of Series ◽  
The One ◽  
Form Of Expression

In the Transactions of the Royal Society for 1807, Dr. Brinkley has investigated the general value of the coefficient of any term in the development of the function( t / e t -1) n , and his result is remarkable for the mode of its expression in terms of the successive differences of the powers of zero, or of the numbers comprised in the general expression ∆ m 0 n . Since that time, in my paper published in the Transactions of the Society for 1815, “On the Development of Exponential Functions,” I have exhibited other, and much more simple as well as more easily calculable expressions for the same coefficient, by means of the same useful and valuable differences, and in that and other subsequent memoirs, have extended their application to a variety of interesting inquiries in the theory of differences and series. It is singular, however, that up to the present time it has never been shown that the formulæ of Dr. Brinkley, and my own, though affording in all cases coincident numerical results, are analytically reconcileable with each other; nor indeed is it at all easy to see either from the course of his investigation, which turns upon an intricate application of the combinatory analysis, or from the nature of the formula itself, how it is possible to pass from the one form of expression to the other so as to show their identity. This is what I now propose. Referring to my “Collection of Examples in the Calculus of Finite Differences,” will be found the following relation, which enables us to pass from the differences of any one power of zero, as 0 z , to those of any other, as 0 x + n , viz.— {log (1 + ∆) } n . f (∆)0 x = x ( x - 1) .... ( x - n +1). f (∆)0 x - n , or changing x into x + n , {log (1 + ∆)} n . f (∆)0 x + n = ( x +1)( x +2) .... ( x + n ). f (∆)0 x .

scholarly journals ANALYSIS OF AN ANTI-PARALLEL MEMRISTOR CIRCUIT

2018 ◽  
Vol 8 (2) ◽  
pp. 9-14
Author(s):  
Valeri Mladenov ◽  
Stoyan Kirilov
Keyword(s):  
Boundary Condition ◽  
Computer Analysis ◽  
Finite Differences ◽  
The State ◽  
The Other ◽  
State Variables ◽  
Initial Values ◽  
Finite Differences Method ◽  
Basic Purpose ◽  
Supply Current

The basic purpose of the present paper is to propose an extended investigation and computer analysis of an anti-parallel memristor circuit with two equivalent memristor elements with different initial values of the state variables using a modified Boundary Condition Memristor (BCM) Model and the finite differences method. The memristor circuit is investigated for sinusoidal supply current at different magnitudes – for soft-switching and hard-switching modes, respectively. The influence of the initial values of the state variables on the circuit’s behaviour is presented as well. The equivalent i-v and memristance-flux and the other important relationshipsof the memristor circuit are also analyzed.

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