Relations, Functions, Partial Functions

2011 ◽  
pp. 101-164
Author(s):  
Jean Gallier
Keyword(s):  
Partial Functions

Methods and clinical applications in nuclear cardiology: a position statement

2002 ◽  
Vol 41 (01) ◽  
pp. 3-13 ◽  
Author(s):  
M. Schäfers
Keyword(s):  
Nuclear Cardiology ◽  
Clinical Medicine ◽  
Imaging Techniques ◽  
Cost Savings ◽  
Cost Benefit ◽  
Position Statement ◽  
Electron Beam Tomography ◽  
Partial Functions ◽  
Cost Benefit Ratio ◽  
Medicine Today

SummaryNuclear cardiological procedures have paved the way for non-invasive diagnostics of various partial functions of the heart. Many of these functions cannot be visualised for diagnosis by any other method (e. g. innervation). These techniques supplement morphological diagnosis with regard to treatment planning and monitoring. Furthermore, they possess considerable prognostic relevance, an increasingly important issue in clinical medicine today, not least in view of the cost-benefit ratio.Our current understanding shows that effective, targeted nuclear cardiology diagnosis – in particular for high-risk patients – can contribute toward cost savings while improving the quality of diagnostic and therapeutic measures.In the future, nuclear cardiology will have to withstand mounting competition from other imaging techniques (magnetic resonance imaging, electron beam tomography, multislice computed tomography). The continuing development of these methods increasingly enables measurement of functional aspects of the heart. Nuclear radiology methods will probably develop in the direction of molecular imaging.

On the action of the implicative closure operator on the set of partial functions of the multivalued logic

2021 ◽  
Vol 31 (3) ◽  
pp. 155-164
Author(s):  
Sergey S. Marchenkov
Keyword(s):  
Closure Operator ◽  
Multivalued Logic ◽  
Partial Functions ◽  
Logical Implication

Abstract On the set P k ∗ $\begin{array}{} \displaystyle P_k^* \end{array}$ of partial functions of the k-valued logic, we consider the implicative closure operator, which is the extension of the parametric closure operator via the logical implication. It is proved that, for any k ⩾ 2, the number of implicative closed classes in P k ∗ $\begin{array}{} \displaystyle P_k^* \end{array}$ is finite. For any k ⩾ 2, in P k ∗ $\begin{array}{} \displaystyle P_k^* \end{array}$ two series of implicative closed classes are defined. We show that these two series exhaust all implicative precomplete classes. We also identify all 8 atoms of the lattice of implicative closed classes in P 3 ∗ $\begin{array}{} \displaystyle P_3^* \end{array}$ .

scholarly journals Quotienting the delay monad by weak bisimilarity

2017 ◽  
Vol 29 (1) ◽  
pp. 67-92 ◽  
Author(s):  
JAMES CHAPMAN ◽  
TARMO UUSTALU ◽  
NICCOLÒ VELTRI
Keyword(s):  
Computer Science ◽  
Equivalence Relation ◽  
Type Theory ◽  
Partial Functions ◽  
Homotopy Type Theory ◽  
Inductive Type ◽  
Alternative Approach ◽  
The One ◽  
Axiom Of Countable Choice

The delay datatype was introduced by Capretta (Logical Methods in Computer Science, 1(2), article 1, 2005) as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. The delay datatype is a monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay datatype quotiented by weak bisimilarity is still a monad–a constructive alternative to the maybe monad. In this paper, we consider the alternative approach of Hofmann (Extensional Constructs in Intensional Type Theory, Springer, London, 1997) of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. With the aid of these principles, we also prove that the quotiented delay datatype delivers free ω-complete pointed partial orders (ωcppos).Altenkirch et al. (Lecture Notes in Computer Science, vol. 10203, Springer, Heidelberg, 534–549, 2017) demonstrated that, in homotopy type theory, a certain higher inductive–inductive type is the free ωcppo on a type X essentially by definition; this allowed them to obtain a monad of free ωcppos without recourse to a choice principle. We notice that, by a similar construction, a simpler ordinary higher inductive type gives the free countably complete join semilattice on the unit type 1. This type suffices for constructing a monad, which is isomorphic to the one of Altenkirch et al. We have fully formalized our results in the Agda dependently typed programming language.

scholarly journals Simultaneously vanishing higher derived limits

2021 ◽  
Vol 9 ◽  
Author(s):  
Jeffrey Bergfalk ◽  
Chris Lambie-Hanson
Keyword(s):  
Euclidean Space ◽  
Metric Spaces ◽  
Inverse System ◽  
Finite Support ◽  
Partial Functions ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Strong Homology ◽  
Finite Support Iteration ◽  
Weakly Compact Cardinal

Abstract In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system $\textbf {A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim ^n\textbf {A}$ (the nth derived limit of $\textbf {A}$ ) vanishes for every $n>0$ . Since that time, the question of whether it is consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for every $n>0$ has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for all $n>0$ . We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to $\lim ^n\textbf {A}=0$ will hold for each $n>0$ . This condition is of interest in its own right; namely, it is the triviality of every coherent n-dimensional family of certain specified sorts of partial functions $\mathbb {N}^2\to \mathbb {Z}$ which are indexed in turn by n-tuples of functions $f:\mathbb {N}\to \mathbb {N}$ . The triviality and coherence in question here generalise the classical and well-studied case of $n=1$ .

Arterial hypertension in patients with active crystal formation

1999 ◽  
Vol 80 (6) ◽  
pp. 415-416
Author(s):  
A. N. Maksudova ◽  
I. G. Salihov ◽  
O. N. Sigitova
Keyword(s):  
Arterial Hypertension ◽  
Interstitial Nephritis ◽  
Purine Metabolism ◽  
Crystal Formation ◽  
Partial Functions ◽  
Ultrasonic Examination ◽  
Active Crystal ◽  
Fast Development

The results of examination of 70 patients with urinary syndrome as oxaliccalcic and uratic crystalluria and in the projection of pelvic system by ultrasonic examination data were analyzed. The study of partial functions of kidneys, purine and oxalic metabolism was performed to estimate hypertension syndrome in patients with dysmetabolic disorders. The comparison of patients with arterial hypertension (15) and normotonia (55) showed the changes in the first group as fast development of dysmetabolic disorders and significant disorder of purine metabolism. The data obtained show the relation between hypertension and interstitial nephritis activity.

scholarly journals Realization of CyberRail Partial Functions in Actual Environment

2005 ◽  
Vol 46 (1) ◽  
pp. 18-22 ◽  
Author(s):  
Takahiko OGINO ◽  
Ryuji TSUCHIYA
Keyword(s):  
Partial Functions ◽  
Actual Environment

scholarly journals Using desirability functions when making managerial decisions in the mineral resource complex

2020 ◽  
Vol 1 (2) ◽  
pp. 218-224
Author(s):  
Artem Valer’evich PONOMAREV ◽  
◽  
Lyudmila Vital’evna VLASOVA ◽  
Irina Vladislavovna PEREGON ◽  
◽  
...  
Keyword(s):  
Computer Technology ◽  
Desirability Function ◽  
Geometric Mean ◽  
Partial Function ◽  
Generalized Function ◽  
Partial Functions ◽  
Desirability Functions ◽  
Managerial Decisions ◽  
First Case ◽  
Business Entities

Relevance. The ever-growing flows of information, tight deadlines for decision-making, the need to maintain competitive positions require ranking calculations to identify leaders and promote leadership factors, ranking units of business entities according to selected criteria, etc. Reliability, visualization, and often acceleration of obtaining results predetermine the appeal to mathematical techniques. Purpose of the study: substantiation of choice of the most acceptable option of the desirability function for use in the process of making managerial decisions related to the development of subsoil resources. Research method: comparative analysis of options for calculating the desirability function, comparison method, methods of mathematical statistics. Results. Two options for calculating the desirability function are considered. The first one developed by E. Harrington, who proposed a special verbal-numerical scale, which made it possible to formalize the system of preferences of different experts. The partial function of desirability, in this case, is constructed so that it is close to "satisfactory" linear one. According to recommendations, the lower and upper boundaries of “satisfactorily” correspond to the minimum and maximum values of indicators for the available data array. There is a slightly different approach to the determination of the partial function of desirability described in a number of works. The generalized desirability function is a convolution of particular functions and is defined as the geometric mean or logarithmic mean. It is possible to use weighting factors with different significance of particular desirability functions. Comparison of various approaches is made for the conditions of management of logistics and equipment of OOO Gazpromtransgaz-Yekaterinburg. An analysis of the results shows that despite discreteness of the particular desirability functions, which in the first case lie in the range from 0.37 to 0.69, and give the entire range of values from 0 to 1 in the second one, the ratings for the generalized desirability function completely coincide with regard to determining a rating of quarterly values of industrial and economic activity. It follows that both methods of calculating partial functions are legitimate, however, the first option seems more convenient technically due to the absence of values of partial desirability functions close to zero, which complicate calculations using computer technology. A third version of the rating was also tested using calculated percentage estimates of the indicators under consideration with respect to the maximum desired value (taking into account their conversion into the relative indices). As in previous cases, the rating of countries according to the generalized function of desirability turned out to be comparable with previously calculated. Conclusions. Comparative calculations confirmed the validity of both options for finding particular desirability functions, since the ratings for the generalized desirability function coincide. The first version of the calculation is more convenient due to the lack of values of particular desirability functions close to zero, which simplifies the use of computer technology

scholarly journals Convergence Classes and Spaces of Partial Functions

2003 ◽  
pp. 75-115
Author(s):  
Anthony Karel Seda ◽  
Roland Heinze ◽  
Pascal Hitzler
Keyword(s):  
Partial Functions
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