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Title: short introduction to nonstandard analysis.

Abstract: These lecture notes, to be completed in a later version, offer a short and rigorous introduction to Nostandard Analysis, mainly aimed to reach to a presentation of the basics of Loeb integration, and in particular, Loeb measures. The Abraham Robinson version of Nostandard Analysis is pursued, with a respective incursion into Superstructures. Two formal languages are used, one simpler at first, and then later, one for the full blown theory. A number of detailed comments, not often found in textbooks, accompanies the text.

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## Nonstandard Analysis

Nonstandard analysis is a branch of mathematical logic which introduces hyperreal numbers to allow for the existence of "genuine infinitesimals," which are numbers that are less than 1/2, 1/3, 1/4, 1/5, ..., but greater than 0. Abraham Robinson developed nonstandard analysis in the 1960s. The theory has since been investigated for its own sake and has been applied in areas such as Banach spaces , differential equations, probability theory, mathematical economics, and mathematical physics.

Loosely, nonstandard methods replace higher-order concepts with first-order analogs. It looks at them from a different angle. Crucially, however, the angle at which the nonstandard analyst looks at the axioms of analysis provides for an average case reduction in complexity that provides shorter proofs of various results, and will one day lead to the proof of a result which is not accessible to classical mathematics without nonstandard methods, precisely because its classical proof is too long to write down in the length of time humans will reside on Earth.

In addition, in the nonstandard analysis community, there is a growing number of results that are not being translated into standard results, because the intuitive content of certain theorems is greater and/or clearer when left in nonstandard terminology. Examples include the use of nonstandard analysis in mathematical economics to describe the behavior of large economies and the use of nonstandard methods to give meaning to concepts that do not classically make sense, such as certain products of infinitely many independent, equally weighted random variables.

Portions of this entry contributed by Matt Insall ( author's link )

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## Non-standard analysis

2020 Mathematics Subject Classification: Primary: 26E35 Secondary: 03H05 [ MSN ][ ZBL ]

A branch of mathematical logic concerned with the application of the theory of non-standard models to investigations in traditional domains of mathematics: mathematical analysis, function theory, the theory of differential equations, probability theory, and others. The basic method of non-standard analysis can roughly be described as follows. One considers a certain mathematical structure $M$ and constructs a first-order logico-mathematical language that reflects those aspects of this structure that are of interest to the investigator. Then one constructs by methods of model theory a non-standard model of the theory of $M$ that is a proper extension of $M$. Under a suitable construction new, non-standard, elements of the model can be interpreted as limiting "ideal" elements of the original structure. For example, if as the original structure one takes the field of real numbers, then it is natural to treat the non-standard elements of the model as "infinitesimals" , that is, as infinitely large or infinitely small, but non-zero, real numbers. Then all the usual relations between real numbers carry over to the non-standard elements, with the preservation of all their properties that can be expressed in the logico-mathematical language. Similarly, in the theory of filters on a given set the intersection of all non-empty elements of the filter determines a non-standard element; in topology this gives rise to a family of non-standard points situated "infinitely close" to a given point. The interpretation of the non-standard elements of a model often makes it possible to give convenient criteria for ordinary concepts in terms of non-standard elements. For example, it can be proved that a standard real-valued function $f$ is continuous at a standard point $x_0$ if and only if $f(x)$ is infinitely close to $f(x_0)$ for all (non-standard) points $x$ infinitely close to $x_0$. The criterion thus obtained can be successfully applied to proofs of ordinary mathematical results.

Naturally, results obtained by methods of non-standard analysis can be, in principle, proved in the standard theory, but the consideration of a non-standard model has the distinct advantage that it allows one to actually introduce into the argument "ideal" elements, making it possible to give lucid statements for many concepts connected with a limit transition from the finite to the infinite. Non-standard analysis places the ideas of G. Leibniz and his followers, about the existence of infinitely small non-zero quantities, on a strict mathematical basis, a circle of ideas (the infinitesimal calculus ) which in the subsequent development of mathematical analysis was rejected in favour of the precise concept of the limit of a variable quantity.

A number of new facts have been discovered by means of non-standard analysis. Many classical proofs gain substantially in clarity when presented by means of non-standard analysis. Non-standard analysis has been used successfully in constructing a rigorous theory of certain semi-empirical methods of mechanics and physics.

In recent years numerous developments involving non-standard analysis, especially in stochastic analysis, the theory of dynamical systems and mathematical physics, have taken place. The references below cover such aspects.

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## nLab nonstandard analysis

analysis ( differential / integral calculus , functional analysis , topology )

epsilontic analysis

infinitesimal analysis

computable analysis

Introduction

## Basic concepts

triangle inequality

metric space , normed vector space

open ball , open subset , neighbourhood

metric topology

sequence , net

convergence , limit of a sequence

compactness , sequential compactness

differentiation , integration

topological vector space

## Basic facts

continuous metric space valued function on compact metric space is uniformly continuous

intermediate value theorem

extreme value theorem

Heine-Borel theorem

## Non-standard Analysis

- Abraham Robinson

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Considered by many to be Abraham Robinson’s magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson’s attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for anyone interested in non-standard analysis. It treats in rich detail many areas of application, including topology, functions of a real variable, functions of a complex variable, and normed linear spaces, together with problems of boundary layer flow of viscous fluids and rederivations of Saint-Venant’s hypothesis concerning the distribution of stresses in an elastic body.

"Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and application of non-standard analysis which grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus."— L'Enseignement Mathématique

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June 1, 1972

## Nonstandard Analysis

By Martin Davis & Reuben Hersh

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The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis [1] [2] [3] instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.

Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson . [4] [5] He wrote:

... the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection ... that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latter

Robinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle . Robinson continued:

However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits. [6]

Robinson continues:

It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory .

In 1973, intuitionist Arend Heyting praised non-standard analysis as "a standard model of important mathematical research". [7]

- 1 Introduction
- 2 Basic definitions
- 3.1 Historical
- 3.2 Pedagogical
- 3.3 Technical
- 4 Approaches to non-standard analysis
- 5 Robinson's book
- 6 Invariant subspace problem
- 7.1 Applications to calculus
- 9 Logical framework
- 10 Internal sets
- 11 First consequences
- 12 κ -saturation
- 13 See also
- 14 Further reading
- 15 References
- 16 Bibliography
- 17 External links

## Introduction

A non-zero element of an ordered field [math]\displaystyle{ \mathbb F }[/math] is infinitesimal if and only if its absolute value is smaller than any element of [math]\displaystyle{ \mathbb F }[/math] of the form [math]\displaystyle{ \frac{1}{n} }[/math] , for [math]\displaystyle{ n }[/math] a standard natural number. Ordered fields that have infinitesimal elements are also called non-Archimedean . More generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle . A field that satisfies the transfer principle for real numbers is a hyperreal field, and non-standard real analysis uses these fields as non-standard models of the real numbers.

Robinson's original approach was based on these non-standard models of the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966 and is still in print. [8] On page 88, Robinson writes:

The existence of non-standard models of arithmetic was discovered by Thoralf Skolem (1934). Skolem's method foreshadows the ultrapower construction [...]

Several technical issues must be addressed to develop a calculus of infinitesimals. For example, it is not enough to construct an ordered field with infinitesimals. See the article on hyperreal numbers for a discussion of some of the relevant ideas.

## Basic definitions

In this section we outline one of the simplest approaches to defining a hyperreal field [math]\displaystyle{ ^*\mathbb{R} }[/math] . Let [math]\displaystyle{ \mathbb{R} }[/math] be the field of real numbers, and let [math]\displaystyle{ \mathbb{N} }[/math] be the semiring of natural numbers. Denote by [math]\displaystyle{ \mathbb{R}^{\mathbb{N}} }[/math] the set of sequences of real numbers. A field [math]\displaystyle{ ^*\mathbb{R} }[/math] is defined as a suitable quotient of [math]\displaystyle{ \mathbb{R}^\mathbb{N} }[/math] , as follows. Take a nonprincipal ultrafilter [math]\displaystyle{ F \subseteq P(\mathbb{N}) }[/math] . In particular, [math]\displaystyle{ F }[/math] contains the Fréchet filter . Consider a pair of sequences

We say that [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] are equivalent if they coincide on a set of indices that is a member of the ultrafilter, or in formulas:

The quotient of [math]\displaystyle{ \mathbb{R}^\mathbb{N} }[/math] by the resulting equivalence relation is a hyperreal field [math]\displaystyle{ ^*\mathbb{R} }[/math] , a situation summarized by the formula [math]\displaystyle{ ^*\mathbb{R} = {\mathbb{R}^\mathbb{N}}/{F} }[/math] .

There are at least three reasons to consider non-standard analysis: historical, pedagogical, and technical.

Much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity . As noted in the article on hyperreal numbers , these formulations were widely criticized by George Berkeley and others. It was a challenge to develop a consistent theory of analysis using infinitesimals and the first person to do this in a satisfactory way was Abraham Robinson. [6]

In 1958 Curt Schmieden and Detlef Laugwitz published an Article "Eine Erweiterung der Infinitesimalrechnung" [9] - "An Extension of Infinitesimal Calculus", which proposed a construction of a ring containing infinitesimals. The ring was constructed from sequences of real numbers. Two sequences were considered equivalent if they differed only in a finite number of elements. Arithmetic operations were defined elementwise. However, the ring constructed in this way contains zero divisors and thus cannot be a field.

## Pedagogical

H. Jerome Keisler, David Tall, and other educators maintain that the use of infinitesimals is more intuitive and more easily grasped by students than the "epsilon–delta" approach to analytic concepts. [10] This approach can sometimes provide easier proofs of results than the corresponding epsilon–delta formulation of the proof. Much of the simplification comes from applying very easy rules of nonstandard arithmetic, as follows:

together with the transfer principle mentioned below.

Another pedagogical application of non-standard analysis is Edward Nelson 's treatment of the theory of stochastic processes. [11]

Some recent work has been done in analysis using concepts from non-standard analysis, particularly in investigating limiting processes of statistics and mathematical physics. Sergio Albeverio et al. [12] discuss some of these applications.

## Approaches to non-standard analysis

There are two very different approaches to non-standard analysis: the semantic or model-theoretic approach and the syntactic approach. Both these approaches apply to other areas of mathematics beyond analysis, including number theory, algebra and topology.

Robinson's original formulation of non-standard analysis falls into the category of the semantic approach . As developed by him in his papers, it is based on studying models (in particular saturated models ) of a theory . Since Robinson's work first appeared, a simpler semantic approach (due to Elias Zakon) has been developed using purely set-theoretic objects called superstructures . In this approach a model of a theory is replaced by an object called a superstructure V ( S ) over a set S . Starting from a superstructure V ( S ) one constructs another object * V ( S ) using the ultrapower construction together with a mapping V ( S ) → * V ( S ) that satisfies the transfer principle . The map * relates formal properties of V ( S ) and * V ( S ) . Moreover, it is possible to consider a simpler form of saturation called countable saturation. This simplified approach is also more suitable for use by mathematicians who are not specialists in model theory or logic.

The syntactic approach requires much less logic and model theory to understand and use. This approach was developed in the mid-1970s by the mathematician Edward Nelson . Nelson introduced an entirely axiomatic formulation of non-standard analysis that he called internal set theory (IST). [13] IST is an extension of Zermelo–Fraenkel set theory (ZF) in that alongside the basic binary membership relation ∈, it introduces a new unary predicate standard , which can be applied to elements of the mathematical universe together with some axioms for reasoning with this new predicate.

Syntactic non-standard analysis requires a great deal of care in applying the principle of set formation (formally known as the axiom of comprehension), which mathematicians usually take for granted. As Nelson points out, a fallacy in reasoning in IST is that of illegal set formation . For instance, there is no set in IST whose elements are precisely the standard integers (here standard is understood in the sense of the new predicate). To avoid illegal set formation, one must only use predicates of ZFC to define subsets. [13]

Another example of the syntactic approach is the Alternative Set Theory [14] introduced by Petr Vopěnka , trying to find set-theory axioms more compatible with the non-standard analysis than the axioms of ZF.

## Robinson's book

Abraham Robinson's book Non-standard analysis was published in 1966. Some of the topics developed in the book were already present in his 1961 article by the same title (Robinson 1961). [15] In addition to containing the first full treatment of non-standard analysis, the book contains a detailed historical section where Robinson challenges some of the received opinions on the history of mathematics based on the pre–non-standard analysis perception of infinitesimals as inconsistent entities. Thus, Robinson challenges the idea that Augustin-Louis Cauchy 's "sum theorem" in Cours d'Analyse concerning the convergence of a series of continuous functions was incorrect, and proposes an infinitesimal-based interpretation of its hypothesis that results in a correct theorem.

## Invariant subspace problem

Abraham Robinson and Allen Bernstein used non-standard analysis to prove that every polynomially compact linear operator on a Hilbert space has an invariant subspace . [16]

Given an operator T on Hilbert space H , consider the orbit of a point v in H under the iterates of T . Applying Gram–Schmidt one obtains an orthonormal basis ( e i ) for H . Let ( H i ) be the corresponding nested sequence of "coordinate" subspaces of H . The matrix a i,j expressing T with respect to ( e i ) is almost upper triangular, in the sense that the coefficients a i +1, i are the only nonzero sub-diagonal coefficients. Bernstein and Robinson show that if T is polynomially compact, then there is a hyperfinite index w such that the matrix coefficient a w +1, w is infinitesimal. Next, consider the subspace H w of * H . If y in H w has finite norm, then T ( y ) is infinitely close to H w .

Now let T w be the operator [math]\displaystyle{ P_w \circ T }[/math] acting on H w , where P w is the orthogonal projection to H w . Denote by q the polynomial such that q ( T ) is compact. The subspace H w is internal of hyperfinite dimension. By transferring upper triangularisation of operators of finite-dimensional complex vector space, there is an internal orthonormal Hilbert space basis ( e k ) for H w where k runs from 1 to w , such that each of the corresponding k -dimensional subspaces E k is T -invariant. Denote by Π k the projection to the subspace E k . For a nonzero vector x of finite norm in H , one can assume that q ( T )( x ) is nonzero, or | q ( T )( x )| > 1 to fix ideas. Since q ( T ) is a compact operator, ( q ( T w ))( x ) is infinitely close to q ( T )( x ) and therefore one has also | q ( T w )( x )| > 1 . Now let j be the greatest index such that [math]\displaystyle{ |q(T_w) \left (\Pi_j(x) \right)|\lt \tfrac{1}{2} }[/math] . Then the space of all standard elements infinitely close to E j is the desired invariant subspace.

Upon reading a preprint of the Bernstein and Robinson paper, Paul Halmos reinterpreted their proof using standard techniques. [17] Both papers appeared back-to-back in the same issue of the Pacific Journal of Mathematics . Some of the ideas used in Halmos' proof reappeared many years later in Halmos' own work on quasi-triangular operators.

## Other applications

Other results were received along the line of reinterpreting or reproving previously known results. Of particular interest is Teturo Kamae's proof [18] of the individual ergodic theorem or L. van den Dries and Alex Wilkie's treatment [19] of Gromov's theorem on groups of polynomial growth . Nonstandard analysis was used by Larry Manevitz and Shmuel Weinberger to prove a result in algebraic topology. [20]

The real contributions of non-standard analysis lie however in the concepts and theorems that utilize the new extended language of non-standard set theory. Among the list of new applications in mathematics there are new approaches to probability, [11] hydrodynamics, [21] measure theory, [22] nonsmooth and harmonic analysis, [23] etc.

There are also applications of non-standard analysis to the theory of stochastic processes, particularly constructions of Brownian motion as random walks . Albeverio et al. [12] have an excellent introduction to this area of research.

## Applications to calculus

As an application to mathematical education, H. Jerome Keisler wrote An Infinitesimal Approach . [10] Covering non-standard calculus , it develops differential and integral calculus using the hyperreal numbers, which include infinitesimal elements. These applications of non-standard analysis depend on the existence of the standard part of a finite hyperreal r . The standard part of r , denoted st( r ) , is a standard real number infinitely close to r . One of the visualization devices Keisler uses is that of an imaginary infinite-magnification microscope to distinguish points infinitely close together. Keisler's book is now out of print, but is freely available from his website; see references below.

Despite the elegance and appeal of some aspects of non-standard analysis, criticisms have been voiced, as well, such as those by Errett Bishop , Alain Connes , and P. Halmos, as documented at criticism of non-standard analysis .

## Logical framework

Given any set S , the superstructure over a set S is the set V ( S ) defined by the conditions

Thus the superstructure over S is obtained by starting from S and iterating the operation of adjoining the power set of S and taking the union of the resulting sequence. The superstructure over the real numbers includes a wealth of mathematical structures: For instance, it contains isomorphic copies of all separable metric spaces and metrizable topological vector spaces. Virtually all of mathematics that interests an analyst goes on within V ( R ) .

The working view of nonstandard analysis is a set * R and a mapping * : V ( R ) → V (* R ) that satisfies some additional properties. To formulate these principles we first state some definitions.

A formula has bounded quantification if and only if the only quantifiers that occur in the formula have range restricted over sets, that is are all of the form:

For example, the formula

has bounded quantification, the universally quantified variable x ranges over A , the existentially quantified variable y ranges over the powerset of B . On the other hand,

does not have bounded quantification because the quantification of y is unrestricted.

## Internal sets

A set x is internal if and only if x is an element of * A for some element A of V ( R ) . * A itself is internal if A belongs to V ( R ) .

We now formulate the basic logical framework of nonstandard analysis:

- Extension principle: The mapping * is the identity on R .
- Transfer principle : For any formula P ( x 1 , ..., x n ) with bounded quantification and with free variables x 1 , ..., x n , and for any elements A 1 , ..., A n of V ( R ) , the following equivalence holds:
- Countable saturation : If { A k } k ∈ N is a decreasing sequence of nonempty internal sets, with k ranging over the natural numbers, then

One can show using ultraproducts that such a map * exists. Elements of V ( R ) are called standard . Elements of * R are called hyperreal numbers .

## First consequences

The symbol * N denotes the nonstandard natural numbers. By the extension principle, this is a superset of N . The set * N − N is nonempty. To see this, apply countable saturation to the sequence of internal sets

The sequence { A n } n ∈ N has a nonempty intersection, proving the result.

We begin with some definitions: Hyperreals r , s are infinitely close if and only if

A hyperreal r is infinitesimal if and only if it is infinitely close to 0. For example, if n is a hyperinteger , i.e. an element of * N − N , then 1/ n is an infinitesimal. A hyperreal r is limited (or finite ) if and only if its absolute value is dominated by (less than) a standard integer. The limited hyperreals form a subring of * R containing the reals. In this ring, the infinitesimal hyperreals are an ideal .

The set of limited hyperreals or the set of infinitesimal hyperreals are external subsets of V (* R ) ; what this means in practice is that bounded quantification, where the bound is an internal set, never ranges over these sets.

Example : The plane ( x , y ) with x and y ranging over * R is internal, and is a model of plane Euclidean geometry. The plane with x and y restricted to limited values (analogous to the Dehn plane) is external, and in this limited plane the parallel postulate is violated. For example, any line passing through the point (0, 1) on the y -axis and having infinitesimal slope is parallel to the x -axis.

Theorem. For any limited hyperreal r there is a unique standard real denoted st( r ) infinitely close to r . The mapping st is a ring homomorphism from the ring of limited hyperreals to R .

The mapping st is also external.

One way of thinking of the standard part of a hyperreal, is in terms of Dedekind cuts ; any limited hyperreal s defines a cut by considering the pair of sets ( L , U ) where L is the set of standard rationals a less than s and U is the set of standard rationals b greater than s . The real number corresponding to ( L , U ) can be seen to satisfy the condition of being the standard part of s .

One intuitive characterization of continuity is as follows:

Theorem. A real-valued function f on the interval [ a , b ] is continuous if and only if for every hyperreal x in the interval *[ a , b ] , we have: * f ( x ) ≅ * f (st( x )) .

(see microcontinuity for more details). Similarly,

Theorem. A real-valued function f is differentiable at the real value x if and only if for every infinitesimal hyperreal number h , the value

exists and is independent of h . In this case f ′( x ) is a real number and is the derivative of f at x .

## κ -saturation

It is possible to "improve" the saturation by allowing collections of higher cardinality to be intersected. A model is κ - saturated if whenever [math]\displaystyle{ \{A_i\}_{i \in I} }[/math] is a collection of internal sets with the finite intersection property and [math]\displaystyle{ |I|\leq\kappa }[/math] ,

This is useful, for instance, in a topological space X , where we may want |2 X | -saturation to ensure the intersection of a standard neighborhood base is nonempty. [24]

For any cardinal κ , a κ -saturated extension can be constructed. [25]

- Non-standard calculus
- Transfer principle
- Internal set theory
- An Infinitesimal Approach
- Hyperreal number
- Hyperinteger
- Infinitesimal
- Surreal number
- Non-classical analysis
- Smooth infinitesimal analysis
- Criticism of non-standard analysis
- Influence of non-standard analysis
- Hyperfinite set
- Constructive non-standard analysis
- Calculus Made Easy

## Further reading

- E. E. Rosinger, [math/0407178]. Short introduction to Nonstandard Analysis . arxiv.org.
- ↑ Nonstandard Analysis in Practice. Edited by Francine Diener, Marc Diener. Springer, 1995.
- ↑ Nonstandard Analysis, Axiomatically. By V. Vladimir Grigorevich Kanovei, Michael Reeken. Springer, 2004.
- ↑ Nonstandard Analysis for the Working Mathematician. Edited by Peter A. Loeb , Manfred P. H. Wolff. Springer, 2000.
- ↑ Non-standard Analysis. By Abraham Robinson . Princeton University Press, 1974.
- ↑ Abraham Robinson and Nonstandard Analysis : History, Philosophy, and Foundations of Mathematics. By Joseph W. Dauben. www.mcps.umn.edu.
- ↑ 6.0 6.1 Robinson, A. : Non-standard analysis. North-Holland Publishing Co., Amsterdam 1966.
- ↑ Heijting, A. (1973) "Address to Professor A. Robinson. At the occasion of the Brouwer memorial lecture given by Prof. A.Robinson on the 26th April 1973." Nieuw Arch. Wisk. (3) 21, pp. 134—137.
- ↑ Robinson, Abraham (1996). Non-standard analysis (Revised ed.). Princeton University Press. ISBN 0-691-04490-2 .
- ↑ Curt Schmieden and Detlef Laugwitz: Eine Erweiterung der Infinitesimalrechnung , Mathematische Zeitschrift 69 (1958), 1-39
- ↑ 10.0 10.1 H. Jerome Keisler, An Infinitesimal Approach . First edition 1976; 2nd edition 1986: full text of 2nd edition
- ↑ 11.0 11.1 Edward Nelson: Radically Elementary Probability Theory , Princeton University Press, 1987, full text
- ↑ 12.0 12.1 Sergio Albeverio, Jans Erik Fenstad, Raphael Høegh-Krohn, Tom Lindstrøm: Nonstandard Methods in Stochastic Analysis and Mathematical Physics , Academic Press 1986.
- ↑ 13.0 13.1 Edward Nelson : Internal Set Theory: A New Approach to Nonstandard Analysis , Bulletin of the American Mathematical Society, Vol. 83, Number 6, November 1977. A chapter on internal set theory is available at http://www.math.princeton.edu/~nelson/books/1.pdf
- ↑ Vopěnka, P. Mathematics in the Alternative Set Theory. Teubner, Leipzig, 1979.
- ↑ Robinson, Abraham: 'Non-Standard Analysis', Kon. Nederl. Akad. Wetensch. Amsterdam Proc. AM (=Indag. Math. 23), 1961, 432-440.
- ↑ Allen Bernstein and Abraham Robinson, Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos , Pacific Journal of Mathematics 16:3 (1966) 421-431
- ↑ P. Halmos, Invariant subspaces for Polynomially Compact Operators , Pacific Journal of Mathematics, 16:3 (1966) 433-437.
- ↑ T. Kamae: A simple proof of the ergodic theorem using nonstandard analysis , Israel Journal of Mathematics vol. 42, Number 4, 1982.
- ↑ L. van den Dries and A. J. Wilkie: Gromov's Theorem on Groups of Polynomial Growth and Elementary Logic , Journal of Algebra, Vol 89, 1984.
- ↑ Manevitz, Larry M.; Weinberger, Shmuel: Discrete circle actions: a note using non-standard analysis . Israel J. Math. 94 (1996), 147--155.
- ↑ Capinski M., Cutland N. J. Nonstandard Methods for Stochastic Fluid Mechanics . Singapore etc., World Scientific Publishers (1995)
- ↑ Cutland N. Loeb Measures in Practice: Recent Advances. Berlin etc.: Springer (2001)
- ↑ Gordon E. I., Kutateladze S. S. , and Kusraev A. G. Infinitesimal Analysis Dordrecht, Kluwer Academic Publishers (2002)
- ↑ Salbany, S.; Todorov, T. Nonstandard Analysis in Point-Set Topology . Erwing Schrodinger Institute for Mathematical Physics.
- ↑ Chang, C. C.; Keisler, H. J. Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. xvi+650 pp. ISBN 0-444-88054-2

## Bibliography

- Crowell, Calculus . A text using infinitesimals.
- Hermoso, Nonstandard Analysis and the Hyperreals . A gentle introduction.
- Hurd, A.E. and Loeb, P.A.: An introduction to nonstandard real analysis , London, Academic Press, 1985. ISBN 0-12-362440-1
- Keisler, H. Jerome Elementary Calculus: An Approach Using Infinitesimals . Includes an axiomatic treatment of the hyperreals, and is freely available under a Creative Commons license
- Keisler, H. Jerome: An Infinitesimal Approach to Stochastic Analysis , vol. 297 of Memoirs of the American Mathematical Society, 1984.
- Naranong S., Nonstandard Analysis from a Model-Theoretic Perspective . A streamlined introduction in the spirit of Robinson.
- Robinson, A. Non-standard analysis. Nederl. Akad. Wetensch. Proc. Ser. A 64 = Indag. Math. 23 (1961) 432–440.
- Robert, A. Nonstandard analysis , Wiley, New York 1988. ISBN 0-471-91703-6
- Skolem, Th. (1934) "Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen", Fundamenta Mathematicae 23: 150-161.
- Stroyan, K. A Brief Introduction to Infinitesimal Calculus
- Gordon E., Kusraev A., and Kutateladze S.. Infinitesimal Analysis
- Tao, T. An epsilon of room, II. Pages from year three of a mathematical blog. American Mathematical Society, Providence, RI, 2010 (pp. 209–229).

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## 13. Hong Kong

Human Development Index (2022): 0.952

GDP Per Capita (2024): $72,860

Hong Kong is one of the countries with the highest standard of living, ranked by GDP per capita. Hong Kong has a GDP per capita of $72,860 and a human development index of 0.952.

## 12. Netherlands

Human Development Index (2022): 0.941

GDP Per Capita (2024): $73,320

The Netherlands has a human development index of 0.941 and a GDP per capita of $73,320.

## 11. Denmark

Human Development Index (2022): 0.948

GDP Per Capita (2024): $74,960

Denmark is one of the richest countries in Europe having a GDP per capita of $74,960. Denmark has a human development index of 0.948.

Click to continue reading and see the 10 Countries with Highest Standard of Living Ranked by GDP (PPP) Per Capita .

Suggested Articles:

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Disclosure: None. 30 Countries with Highest Standard of Living Ranked by GDP (PPP) Per Capita is originally published on Insider Monkey.

## EU AI Act: first regulation on artificial intelligence

The use of artificial intelligence in the EU will be regulated by the AI Act, the world’s first comprehensive AI law. Find out how it will protect you.

As part of its digital strategy , the EU wants to regulate artificial intelligence (AI) to ensure better conditions for the development and use of this innovative technology. AI can create many benefits , such as better healthcare; safer and cleaner transport; more efficient manufacturing; and cheaper and more sustainable energy.

In April 2021, the European Commission proposed the first EU regulatory framework for AI. It says that AI systems that can be used in different applications are analysed and classified according to the risk they pose to users. The different risk levels will mean more or less regulation. Once approved, these will be the world’s first rules on AI.

Learn more about what artificial intelligence is and how it is used

## What Parliament wants in AI legislation

Parliament’s priority is to make sure that AI systems used in the EU are safe, transparent, traceable, non-discriminatory and environmentally friendly. AI systems should be overseen by people, rather than by automation, to prevent harmful outcomes.

Parliament also wants to establish a technology-neutral, uniform definition for AI that could be applied to future AI systems.

Learn more about Parliament’s work on AI and its vision for AI’s future

## AI Act: different rules for different risk levels

The new rules establish obligations for providers and users depending on the level of risk from artificial intelligence. While many AI systems pose minimal risk, they need to be assessed.

## Unacceptable risk

Unacceptable risk AI systems are systems considered a threat to people and will be banned. They include:

- Cognitive behavioural manipulation of people or specific vulnerable groups: for example voice-activated toys that encourage dangerous behaviour in children
- Social scoring: classifying people based on behaviour, socio-economic status or personal characteristics
- Biometric identification and categorisation of people
- Real-time and remote biometric identification systems, such as facial recognition

Some exceptions may be allowed for law enforcement purposes. “Real-time” remote biometric identification systems will be allowed in a limited number of serious cases, while “post” remote biometric identification systems, where identification occurs after a significant delay, will be allowed to prosecute serious crimes and only after court approval.

AI systems that negatively affect safety or fundamental rights will be considered high risk and will be divided into two categories:

1) AI systems that are used in products falling under the EU’s product safety legislation . This includes toys, aviation, cars, medical devices and lifts.

2) AI systems falling into specific areas that will have to be registered in an EU database:

- Management and operation of critical infrastructure
- Education and vocational training
- Employment, worker management and access to self-employment
- Access to and enjoyment of essential private services and public services and benefits
- Law enforcement
- Migration, asylum and border control management
- Assistance in legal interpretation and application of the law.

All high-risk AI systems will be assessed before being put on the market and also throughout their lifecycle.

## General purpose and generative AI

Generative AI, like ChatGPT, would have to comply with transparency requirements:

- Disclosing that the content was generated by AI
- Designing the model to prevent it from generating illegal content
- Publishing summaries of copyrighted data used for training

High-impact general-purpose AI models that might pose systemic risk, such as the more advanced AI model GPT-4, would have to undergo thorough evaluations and any serious incidents would have to be reported to the European Commission.

## Limited risk

Limited risk AI systems should comply with minimal transparency requirements that would allow users to make informed decisions. After interacting with the applications, the user can then decide whether they want to continue using it. Users should be made aware when they are interacting with AI. This includes AI systems that generate or manipulate image, audio or video content, for example deepfakes.

On December 9 2023, Parliament reached a provisional agreement with the Council on the AI act . The agreed text will now have to be formally adopted by both Parliament and Council to become EU law. Before all MEPs have their say on the agreement, Parliament’s internal market and civil liberties committees will vote on it.

## More on the EU’s digital measures

- Cryptocurrency dangers and the benefits of EU legislation
- Fighting cybercrime: new EU cybersecurity laws explained
- Boosting data sharing in the EU: what are the benefits?
- EU Digital Markets Act and Digital Services Act
- Five ways the European Parliament wants to protect online gamers
- Artificial Intelligence Act

## Related articles

Benefitting people, the economy and the environment, share this article on:.

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Logic, Foundations of Mathematics, and Computability Theory pp 107–119 Cite as

## Non-Standard Analysis

- W. A. J. Luxemburg 3

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Part of the The University of Western Ontario Series in Philosophy of Science book series (WONS,volume 9)

1. As early as 1934 it was pointed out by Thoralf Skolem (see [17]) that there exist proper extensions of the natural number system which have, in some sense, ‘the same properties’ as the natural numbers. The title of Skolem’s paper indicates that the purpose of it was to show that no axiomatic system specified in a formal language, in Skolem’s case the lower predicate calculus, can characterize the natural numbers categorically. At that time, however, Skolem did not concern himself with the properties of the structures whose existence he had established. In due course these structures became known as non-standard models of arithmetic. For nearly thirty years since the appearance of Skolem’s paper non-standard models were not used or considered in any sense by the working mathematician. Robinson’s fundamental paper, which appeared in 1961 under the title ‘Non-standard Analysis’, (see [11]) changed this situation dramatically. In this paper Abraham Robinson was the first to point out that this highly abstract part of model theory could be applied fruitfully to a theory so far removed from it as the infinitesimal calculus. As a result Robinson obtained a firm foundation for the non-archimedian approach to the calculus based on a number system containing infinitely small and infinitely large numbers, in a manner almost identical to that suggested by Leibniz some three centuries ago, and which predominated the calculus until the middle of the nineteenth century when it was rejected as unsound and replaced by the ϵ, δ-method of Weierstrass.

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- Large Natural Number
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## Bibliography

Bernstein, Allen R. and Robinson, Abraham: 1966, ‘Solution of an Invariant Subspace Problem of K. T. Smith and P. R. Halmos’, Pacific J. Math. 16 , 421–431.

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Loeb, Peter A.: 1971, ‘A Nonstandard Representation of Measurable Spaces and L ∞ ’, Bull. Am. Math. Soc. 77 , 540–544.

Luxemburg, W. A. J.: 1962, Nonstandard Analysis, Lectures on A. Robinson’s Theory of Infinitesimals and Infinitely Large Numbers , Calif. Inst. of Technology, Pasadena 1962 and revised edition, 1964.

Luxemburg, W. A. J. (ed.): 1969, ‘A General Theory of Monads’, in Applications of Model Theory to Algebra, Analysis and Probability (Proc. Symposium on Nonstandard Analysis, Calif. Inst. of Techn., 1967), Holt-Rinehart and Winston, New York, pp. 123–137.

Luxemburg, W. A. J.: 1973, ‘What is Nonstandard Analysis?’, Am. Math. Monthly 80 , 38–67.

Machover, Moshe and Hirschfeld, Jöram: 1969, Lectures on Nonstandard Analysis ( Lecture Notes in Mathematics 94 ), Springer-Verlag, Berlin.

Robinson, Abraham: 1961, ‘Non-Standard Analysis’, Kon. Nederl. Akad. Wetensch. Amsterdam Proc. A64 (= Indag. Math. 23 ), 432–440.

Robinson, Abraham: 1964, ‘On Generalized Limits and Linear Functional’, Pacific J. Math. 14 , 269–283.

Robinson, Abraham: 1966, Non-standard Analysis, Studies in Logic , North-Holland Publ. Co. Amsterdam, second edition, 1975.

Robinson, Abraham: 1967, ‘Non-standard Arithmetic’, Bull. Am. Math. Soc. 73 , 818–843.

Robinson, Abraham: 1973, ‘Nonstandard Points on Algebraic Curves’, J. Number Theory 5 , 301–327.

Robinson, Abraham and Roquette, Peter J.: 1975, ‘On the Finiteness Theorem of Siegel and Mahler Concerning Diophantine Equations’, J. Number Theory 7 , 121–176.

Skolem, Thoralf: 1934, Über die Nicht-charakterisierbarkeit der Zahlenreihe Mittels endlich oder abzahlbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen, Fund. Math. 23 , 150–161.

Stroyan, K. D. and Luxemburg, W. A. J.: 1967, Introduction to the Theory of Infinitesimals , Academic Press, New York.

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Luxemburg, W.A.J. (1977). Non-Standard Analysis. In: Butts, R.E., Hintikka, J. (eds) Logic, Foundations of Mathematics, and Computability Theory. The University of Western Ontario Series in Philosophy of Science, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1138-9_6

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- 2023 PTAB Year in Review: Analysis & Trends: Standard Essential Patents at the PTAB: Are SEPs Faring any Differently than Non-SEPs? Impacts and Analysis

Standard Essential Patents are on the Rise, as is Litigation

Standard-essential patents (SEPs) are on the rise. A key factor undergirding that rise is the desire for device connectivity in all things, and the fact that reliable and robust connectivity is impossible without using key standards that are almost always subject to SEPs. For example, it is estimated that by 2025, more than 26 billion home and workplace devices will be connected to the Internet and have sensors, processors, and embedded software for facilitating connectivity. 2

The economic impact of these connected devices is estimated to be approximately $10 trillion per year by 2025. 3 It is no surprise then that, in the last several years, the number of issued SEPs impacting connectivity has increased dramatically. Just looking at one of the more recent standards—5G cellular communications—the number of declared 5G patent families has increased tenfold between 2017 and 2023, reaching over 60,000. 4 In fact, the number of declared 5G patent families is almost 2.5 times more than the number of patent families declared essential to the previous 4G cellular communication standard. In addition to a surge in quantity, the relevance of SEPs has broadened—wireless and telecom standard technology have become prevalent in everything from biotech and automotive products to home appliances. Consequently, the impact of patents covering standard essential technology is felt, and will continue to be felt, across all major industries.

Predictably, the number of SEPs involved in litigation follows the progression of the technology. With the increased adoption of 4G technology, there was a corresponding rise in litigation of SEPs; the more products that were 4G compliant meant more potential infringers, which led to increased SEP litigation. 5 A similar rise has taken place with the more recent release and increasingly widespread adoption of 5G technology. Unsurprisingly, then, the 4G and 5G standards generally account for more 70% of all SEP litigation. 6

The Threat of Injunctive Relief

As the widespread adoption of standardized technologies continues to rapidly increase, the number of technology implementers that find themselves entangled in SEP disputes will also increase. Technology implementers therefore must be aware of the potential risks involved with SEP litigation. This includes understanding who the SEP holders are, their relative business objectives, and their SEP litigation history. But regardless of the existing SEP landscape, the biggest risk to potential infringers will always be the threat of an injunction.

SEP-based injunctions have not always been viewed as a viable option. SEPs are generally FRAND-encumbered, meaning that the SEP holder has made a promise to license its SEPs on fair reasonable and non-discriminatory terms, which has been viewed by many courts as an admission that monetary damages are adequate compensation. 7 But in 2019, the US Patent and Trademark Office (USPTO), US Department of Justice (DOJ), and National Institute of Standards and Technology (NIST) issued a joint statement to clarify their collective view that SEPs should be eligible for injunctive relief. 8 The statement provided that, as with all other patents, infringement of SEPs should be analyzed for potential injunctive relief under the eBay framework. 9 Then, in June 2022, the DOJ, USPTO, and NIST announced the withdrawal of the 2019 joint statement, and chose not to institute a new SEP policy in its place. This has left the industry without any formal government-sanctioned guidelines for SEP licensing and enforcement. Meanwhile, a number of SEP disputes were brought before the US International Trade Commission (ITC), which led to a string of decisions essentially indicating that SEP-based injunctions (in the form of exclusion orders and/or cease and desist orders) are available at the ITC. 10

With injunctions now a clear possibility, and with the SEP landscape being thrown into a state of flux with both the rollout of the Unitary Patent Court and the European Unions’ Proposed European Commission Regulation For Standard Essential Patents published in April of this year, inter partes reviews (IPRs) offer a strategic option for defendants. A pending or already-instituted IPR decreases a patentee’s chances of obtaining an injunction against a defendant in district court 11 , and increases the likelihood of obtaining a stay of the district court proceedings. Thus, filing an IPR petition early in the course of SEP litigation can be a critical component of the technology implementer’s defense. Moreover, US Patent Trial and Appeal Board (PTAB) judges are generally more receptive to invalidity arguments relating to highly complex technology (which is often the case with SEPs), more so than district court judges and juries, thereby making the PTAB an attractive forum for technology implementers seeking to defend against SEP litigation. 12

For the SEP holder, mitigating the effect of an IPR on a request for injunctive relief should be a primary focus. To this end, SEP holders should research available forums and select an injunction-friendly court if possible (including the, for example, the ITC). SEP holders should also lay out specific details in the complaint to paint the technology implementer as an unwilling licensee (an important factor in determining the availability of injunctive relief involving SEPs), and should seek expedited discovery under FRCP 26(d), which could factor into whether the PTAB decides to use its discretion to deny institution of the IPR.

Petitioners are successfully challenging SEPs at the PTAB

Unsurprisingly, the number of IPRs filed against SEPs has also followed the progression of the technology, and the widespread adoption of agreed-upon standards. As illustrated in Figure 1 below, IPR filings against SEPs saw a spike in 2013-2014, growing to a peak in 2017, before falling to a low in 2019. Then IPR filings against SEPs saw another rise in 2020-2021. These spikes followed the rollouts of 4G and 5G, respectively. The annualized number of SEP IPRs is expected to fall again (as publication), but the rollout and mass incorporation of new connectivity standards (e.g., WiFi 6) will likely cause another spike in SEP litigation and IPRs in the coming months and years.

Petitioners challenging SEPs have had similar success at the PTAB as those challenging non-SEP patents, dispelling any notion that SEPs are necessarily higher quality. As shown in Figure 2 on page 32, IPRs involving electronics-based SEPs have similar institution rates as proceedings involving non-SEP electronics

Figure 1: IPRs Filed Against SEPs

patents. 13 The outlier year, 2020, which saw significantly lower institution rates for IPRs involving electronics-based SEPs coincided with the rollout of the new 5G standard. These lower institution rates are likely due to the unsettled nature of the technology and available universe of prior art.

Additionally, Figure 3 on page 32 shows that IPRs involving electronics-based SEPs have similar claim cancellation rates as proceedings involving non-SEP electronics patents, and actually have higher chances of having all claims cancelled.

One important factor behind the high claim cancellation rates for IPRs involving SEPs—which generally cover highly complex technology with only incremental improvements over existing technology—is the choice of prior art. Seventy-six percent of all IPRs filed against SEPs used non-patent literature (NPLs) as prior art, and 61% of these proceedings specifically used NPLs that were produced explicitly for the purpose of developing and refining standards (SEP NPLs). These include, for example, technical specifications/reports or working group documents produced under the auspices of a standard-setting organization. While the use of NPLs, and specifically SEP NPLs, has led to high claim cancellation rates (76% and 85%, respectively), such references come with their own set of challenges. It can be difficult to prove that these references are printed publications that were publicly accessible sufficiently early, which—despite their compelling substance—has led to relatively low institution rates (51% for NPLs and 57% for SEP NPLs). It is important for petitioners seeking to file SEP IPRs to select counsel familiar with these unique challenges since it is very difficult to cure defective IPR petitions before the PTAB.

Figure 2: Proceeding Institution Rate (Electronics IPRs)

Considerations for Petitioners and Patent Owners

In light of the difficulty in proving that SEP NPL qualifies as prior art, petitioners should consider presenting both a set of patent-based grounds and a set of non-patent-based grounds in a single IPR petition (if possible) challenging an SEP. Doing so may allow petitioners to both avoid the lower institutions rates and take advantage of the higher claim cancellation rates associated with using NPLs as prior art. If it is not possible to fit both sets of grounds in a single petition, then petitioners should consider filing two petitions and highlighting the potential for a public accessibility challenge to the set of non-patent-based grounds as justification for instituting both petitions. At the very least, this approach will increase the likelihood that the SEP holder will raise any public accessibility challenge prior to institution, and may in turn increase the chances that the PTAB will address or resolve these issues at institution.

Additionally, petitioners should engage experts to authenticate these NPL references, and help draw clear lines of correlation between the NPLs and the challenged SEP, which were each drafted for and by different individuals. These experts would preferably have personal experience with the relevant standard setting organizations (SSOs) that produced the SEP NPLs being considered for prior art. This may mean that the petitioner engages multiple experts: one to authenticate and give context to the NPLs and another to speak to patentability, including factors relevant to obviousness and reasons to combine the prior art. This may mean that the petitioner engages multiple experts: one to authenticate and give context to the NPLs and another to speak to patentability, including factors relevant to obviousness and reasons to combine the prior art.

Petitioners should also be aware of possible priority date issues that can impact the available pool of prior art. SEP holders tend to file applications as early as possible as they compete to get their proposed technology adopted as the standard. The earlier the application, the more likely that continuation or divisional applications were filed in an attempt to have these later-filed claim sets read on the final version of the standard. This means that if the SEP being challenged claims priority to an earlier filed application, the claims of the challenged SEP may not be supported by the earlier application(s). This could prevent the patent owner from getting an earlier priority date, thereby increasing the available pool of prior art by a couple months or even years. This can make all the difference when dealing with SEPs that are generally in highly congested technology spaces and may cover only incremental changes.

On the other side, patentees’ strategies should include challenging the public availability of the asserted references at the institution stage. This may include engaging multiple experts as well, where one is specifically tasked with rebutting the documentation and distribution practices of the relevant SSOs. Patentees should also contact the named inventor(s) to get the

Figure 3: Claim Cancellation Outcomes at FWD (Electronics IPRs) 14

complete invention story, including facts relevant to objective indicia evidence. As technology implementers will often argue that SEPs only cover incremental changes to previous versions of a standard, being able to tell a compelling story of why those changes would not in fact have been obvious will be important. Finally, in light of the highly congested technology spaces that SEPs generally cover, patentees should also fully understand art cited and applied during prosecution of the entire SEP family. Additionally, patentees should consider developing a fulsome record during prosecution of the SEPs, including citing all relevant references in an IDS. Patentees should then seek to leverage past precedential decisions to show that art or arguments applied in the IPR are redundant of art or arguments presented during prosecution. 14 Indeed, the PTAB has demonstrated “a commitment to defer to previous Office evaluations of the evidence of record unless material error is shown.” 15

SEPs Moving Forward

IPRs will continue to play a critical role in SEP assertion efforts. The PTAB has become well-versed in dealing with SEP challenges, and in comparison to district court judges and juries, PTAB judges are generally more receptive to complex technical positions and unpatentability arguments. Thus, stakeholders will benefit from incorporating PTAB strategy into their overall SEP assertion strategy.

1. A similar article was published in Sterne Kessler’s PTAB Year in Review 2021. This article has been updated to include new statistics and developments. 2. Ménière Yann, Ilja Rudyk & Javier Valdes, Patents and the Fourth Industrial Revolution: The Inventions Behind Digital Transformation 10 (Eur. Pat. Off. ed., 2017). 3. Communication from the Commission to the European Parliament, the Council and the European Economic and Social Committee Setting Out the EU Approach to Standard Essential Patents, at 1, COM (2017) 712 final (Nov. 29, 2017)(noting the potential is up to EUR 9 trillion per year in developed countries). 4. Tim Pohlmann, Magnus Buggenhagen, & Marco Richter, Who is Leading the 5G Patent Race? (LexisNexis® IPLytics ed., 2023). 5. Report: Litigation Landscape of Standard-Essential Patents 2 (Darts-IP ed., 2019). 6. Tim Pohlmann, How to Navigate Risk Webinar Part 1: The Role of SEPs & Standards in the Auto Industry (IPLytics GmbH ed., 2021). 7. Realtek Semiconductor Corporation v. LSI Corporation and Agere Systems LLC, 946 F. Supp. 2d 998 (N.D. Cal. 2013). 8. U.S. Pat. & Trademark Off., U.S. Dep’t of Just. & Nat’l Inst. of Standards & Tech., Policy Statement on Remedies for Standards-Essential Patents Subject to Voluntary F/RAND Commitments, at 4-5 (Dec. 19, 2019). 9. Id. at 6. 10. See, e.g., Certain Memory Modules and Components Thereof (Inv. No. 337-TA-1089), Certain LTE- and 3G-Compliant Cellular Communications Devices (Inv. No. 337- TA-1138), Certain UMTS and LTE Cellular Communication Modules and Products Containing the Same (Inv. No. 337-TA-1240). 11. See, e.g., DNA Genotek Inc. v. Spectrum Sols. L.L.C., Case No.: 16-CV-1544 JLS (NLS) (S.D. Cal. Aug. 11, 2016) (denying a preliminary injunction for patent infringement based on an IPR filed against the asserted patent); Sciele Pharma Inc. v. Lupin Ltd., 684 F.3d 1253, 1263 (Fed. Cir. 2012) (vacating a preliminary injunction because “the district court incorrectly concluded that [Defendant] failed to raise a substantial question of validity regarding the asserted claims of the [] patent”). 12. Importantly, courts have held that an implementer cannot be criticized for challenging the validity of an SEP, and doing so does not render the implementer an unwilling licensee (a label that in some jurisdictions can increase the likelihood of an injunction). See, e.g., Motorola Mobility LLC, 156 F.T.C. 147, 205-06 (2013). 13. A Docket Navigator search of motion success indicated petitions against non-SEP electronics patents have a 66% institution rate and petitions against electronics-based SEPs have a 59% institution rate. 14. See, e.g. Advanced Bionics , LLC v. MED-EL Elektromedizinische Geräte GmbH, IPR2019-01469 (PTAB Feb. 13, 2020); Oticon Med. AB v. Cochlear Ltd., IPR2019-00975 (PTAB Oct. 16, 2019); Becton, Dickinson & Co. v. B. Braun Melsungen AG, IPR2017-01586 (P.T.A.B. Dec. 15, 2017). 15. Advanced Bionics .

This article appeared in the 2023 PTAB Year in Review: Analysis & Trends report.

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DISCLAIMER: Because of the generality of this update, the information provided herein may not be applicable in all situations and should not be acted upon without specific legal advice based on particular situations.

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## Values of multiparametric and biparametric MRI in diagnosing clinically significant prostate cancer: a multivariate analysis

- Xiao Feng 1 na1 ,
- Xin Chen 2 na1 ,
- Peng Peng 1 ,
- He Zhou 1 ,
- Yi Hong 1 ,
- Chunxia Zhu 1 ,
- Libing Lu 1 ,
- Siyu Xie 1 ,
- Sijun Zhang 1 &
- Liling Long 1

BMC Urology volume 24 , Article number: 40 ( 2024 ) Cite this article

21 Accesses

Metrics details

To investigate the value of semi-quantitative and quantitative parameters (PI-RADS score, T2WI score, ADC, Ktrans, and Kep) based on multiparametric MRI (mpMRI) or biparametric MRI (bpMRI) combined with prostate specific antigen density (PSAD) in detecting clinically significant prostate cancer (csPCa).

A total of 561 patients (276 with csPCa; 285 with non-csPCa) with biopsy-confirmed prostate diseases who underwent preoperative mpMRI were included. Prostate volume was measured for calculation of PSAD. Prostate index lesions were scored on a five-point scale on T2WI images (T2WI score) and mpMRI images (PI-RADS score) according to the PI-RADS v2.1 scoring standard. DWI and DCE-MRI images were processed to measure the quantitative parameters of the index lesion, including ADC, Kep, and Ktrans values. The predictors of csPCa were screened by logistics regression analysis. Predictive models of bpMRI and mpMRI were established. ROC curves were used to evaluate the efficacy of parameters and the model in diagnosing csPCa.

The independent diagnostic accuracy of PSA density, PI-RADS score, T2WI score, ADCrec, Ktrans, and Kep for csPCa were 80.2%, 89.5%, 88.3%, 84.6%, 58.5% and 61.6%, respectively. The diagnostic accuracy of bpMRI T2WI score and ADC value combined with PSAD was higher than that of PI-RADS score. The combination of mpMRI PI‑RADS score, ADC value with PSAD had the highest diagnostic accuracy.

## Conclusions

PI-RADS score according to the PI-RADS v2.1 scoring standard was the most accurate independent diagnostic index. The predictive value of bpMRI model for csPCa was slightly lower than that of mpMRI model, but higher than that of PI-RADS score.

Peer Review reports

Prostate cancer (PCa) is the second most common male cancer with the highest incidence in Western countries [ 1 ]. Multiparametric magnetic resonance imaging (mpMRI) is an efficient non-invasive tool for the diagnosis, staging, and monitoring of PCa [ 2 ]. The prostate imaging reporting and data system (PI-RADS) is a 5-point scale used to predict the possibility of clinically significant prostate cancer (csPCa) based on the findings of mpMRI, which includes T2-weighted imaging (T2WI), diffusion weighted imaging (DWI), and dynamic contrast-enhanced (DCE) [ 3 ]. However, its diagnosis is based on the subjective and semi-quantitative results of mpMRI. The latest PI-RADS v2.1which update of PI-RADS v2.0 in 2019, still does not incorporate clinical data and quantitative parameters, and shows no significant value in DEC imaging. Moreover, the current system does not cover suggestions for PI-RADS category 3 lesions and MRI follow-up [ 3 , 4 ]. Studies have found no significant difference in the diagnostic efficiency for csPCa between biparametric magnetic resonance imaging (bpMRI) and mpMRI [ 5 , 6 ]. Use of the combination of PI-RADS scores, patient’s age, prostate specific antigen (PSA) level, and prostate specific antigen density (PSAD) has been shown to increase the detection rate of csPCa, thus providing a more evaluable reference for clinical decision-making [ 7 , 8 ]. In the present study, the subjects were assigned into csPCa group and non-csPCa group based on the pathological findings. Regarding the limitations of PI-RADSv2.1, we assessed the csPCa-predicting potential of the biparametric and multiparametric models, involving the semi-quantitative and quantitative parameters of mpMRI and bpMRI (e.g., PI-RADS scores and T2 weighted image [T2WI] score according to the latest PI-RADS v2.1 scoring standard, apparent diffusion coefficient [ADC], volume transfer constant between blood plasma and the extracellular extravascular space [K trans ], rate constant between the extracellular extravascular space and the blood plasma [K ep ]) and clinical parameters (PSAD).

## Material and methods

Since 2019, the imaging and clinical data were retrospectively collected from 634 patients who underwent prostate mpMRI at our hospital due to the increase in PSA level, and were confirmed by prostate biopsy or RP(198, 31%) between June 2015 and December 2020. The mpMRI was performed before or four weeks after biopsy to minimize the effect of artefacts induced by blood pooling within the gland. A total of 561 patients (age range 28–92 years; median 67 years) were included after excluding those who had history of treatment, incomplete data, such as PSA without specific value(> 100 ng/ml), or overlapping features with other tumors. Of them, 285 (50.8%) were assigned to the non-csPCa group and 276 (49.2%) to the csPCa group. The study flowchart is presented in Fig. 1 .

Study flowchart shows patient inclusions and exclusions

This study was a retrospective cohort study approved by the institutional review board and complied with HIPAA. The requirement for informed consent was waived off by the human investigation committee at our institution.

## MRI Protocol

MRI was performed with a 3.0-T scanner (Siemens MAGNETOM Verio or Prisma, German) using a 16-channel phased-array body coil. The protocol included axial and sagittal T2WI, axial DWI and DCE in accordance with PI-RADS v2.1. b-values used for DWI included 0, 1000, and 2000 mm/s 2 . ADC map was automatically calculated from b-values of 0 and 1000 mm/s 2 . The MRI scanning parameters are presented in Tables 1 and 2 .

## Image analysis

Prostate volume was measured according to the PI-RADS v2.1 [ 3 ] standard for the calculation of PSAD. Prostate index lesions of each patient were scored on a five-point scale on T2WI images (T2WI score) and mpMRI images (PI-RADS score) by a senior radiologist blinded to the pathological results and having 12 years of MRI experience (having read more than eighty thousand patients’ MRI images, these included about 1,500 prostate MRI), according to the PI-RADS v2.1 scoring standard.

All images were sent to a workstation. Specific software was used to process DW and DCE images (4D-Tissu). A senior radiologist analyzed all the mpMRI images that qualified the inclusion criteria to identify index prostate cancer foci. The mean ADC, Kep, and Ktrans values were evaluated on a selected region of interest encompassing as much of the inner part of the lesion as possible without contacting the edges (Figs. 2 and 3 ).

ADC measurements of prostate cancer. A shows an abnormally high signal focus in the left peripheral zone on the DWI (B-value 1000 mm/s 2 ); B , C show that the mean ADC value of the lesion in the left peripheral zone is 0.670 × 10 -3 mm 2 /s

Parameter diagram of DCE measurements in prostate cancer. A Early dynamic contrast enhanced image shows avid enhancement within the anterior lesion (arrows). B The signal intensity-time curve of ROI3 shows a plateau of rapid rise and slow fall, while ROI1.2.4 shows an inflow curve of slow rise. E The Ktrans and Kep values of the index lesion are 1.110/min and 1.279/min. C , D , F shows the Ktrans and Kep values in the normal area of bilateral transition zone and right peripheral zone

## Reference standard

Each patient underwent both systematic biopsy (with an average of 12 random samples from the entire prostate gland) and target biopsy (with at least three samples obtained from each lesion identified by MRI). Target sampling was performed with an MRI/TRUS fusion, alternately using the cognitive technique or dedicated software, coupled with various commercially-available ultrasound tools. For patients undergoing radical prostatectomy (RP) after puncture, pathological results obtained after RP were used as the gold standard for diagnosis. Post-RP specimens were sectioned at 4–6 mm intervals, and stained with hematoxylin and eosin (H&E). A pathologist recorded the presence or absence of PCa, tumor location, and determined the tumor Gleason score (GS) of biopsy specimens. GS was calculated according to the 2014 International Society of Urological Pathology Modified Gleason Grading System [ 9 ]. The definition of csPCa was a tumor with GS ≥ 7, or GS = 3 + 3 plus tumor size ≥ 0.5 mL [ 10 ]. Tumor size was calculated from mpMR images, most commonly T2WI. When multiple foci of PCa were found, the focus with the highest GS was considered as the index lesion.

## Statistical analysis

According to the pathological results after biopsy, patients were divided into two groups, i.e., csPCa and non-csPCa. Between-group differences with respect to each parameter (PSA density, PI-RADS score, T2WI score, ADC, Ktrans, and Kep) were assessed using the Independent-samples U test. The predictors of csPCa were screened by logistics regression analysis. In case of multicollinearity, logistic regression analysis was performed using the likelihood ratio forward method to screen variables in the model. mpMRI and bpMRI predictive models were established and receiver operating characteristic (ROC) curves were plotted to evaluate the efficiency of each parameter and the model in diagnosing csPCa. The diagnostic performance was compared using the DeLong test. Two-tailed P values < 0.05 were considered indicative of statistical significance.

All the 561 cases were confirmed by biopsies and eligible for this study. The non-csPCa group comprised of 285 patients, including 168 with benign prostatic hyperplasia (BPH), 49 with prostatitis, 20 with prostate intraepithelial neoplasia (PIN), and 48 with clinically insignificant Pca (ciPCa). The CsPCa group comprised of 276 patients, including 69 with International Society of Urological Pathology (ISUP) grade 2, 77 with ISUP grade 3, 60 with ISUP grade 4, and 70 with ISUP grade 5 prostate cancer (Fig. 4 ).

Representative case of prostate cancer An 80-year-old man with PSA 73.2 ng/mL, PSAD 0.77 ng/mL 2 , and Gleason score 3 + 4 prostate cancer confirmed after RP. A , B Axial T2WI and FS-T2WI sequences show a T2 hypointense nodule (arrows) involving the left peripheral zone with extraprostatic extension; T2WI score = 5. C Diffusion-weighted image (b = 2000) shows a markedly hyperintense signal (arrows) corresponding to (A) and (B). D ADC map image shows focal hypointense signal corresponding to (C), ADC value of the lesion is 0.829 × 10 -3 mm 2 /s; DWI PI-RADS = 5. E Early dynamic contrast enhanced image shows avid enhancement within the anterior lesion (arrows), DCE MRI PI-RADS = positive. PI-RADS score = 5. F , G The Ktrans and Kep values of the anterior lesion are 0.345/min and 1.249/min, respectively. H Gross morphology of the RP specimen. (I) Microscopic pathological view of the index lesion

## Normality test of semi-quantitative and quantitative parameters

The one-sample Kolmogorov–Smirnov test showed non-normal distribution of PI-RADS score, T2WI score, ADC, K trans , K ep , PSAD, and patient’s age ( P < 0.001 for all).

## Mann–Whitney U test of semi-quantitative and quantitative parameters

The csPCa group had significantly higher PSAD, PI-RADS score, T2WI score, K trans and K ep , but significantly lower ADC compared to the non-csPCa group ( P < 0.05 for all). No significant difference in age was detected between csPCa and non-csPCa group ( P = 0.099, Table 3 ).

## Univariable logistic regression analysis of semi-quantitative and quantitative parameters and their respective diagnostic efficiency

Univariable logistic regression analysis showed significant differences between csPCa and non-csPCa groups with respect to PI-RADS score, T2WI score, ADC, and PSAD ( P < 0.05 for all), but not with respect to K trans , K ep and ADC reciprocal (ADCrec). Furthermore, receiver operating characteristic (ROC) curve analysis revealed that PI-RADS score showed the highest diagnostic efficiency for csPCa, followed by T2WI score, ADCrec, PSAD, K ep, and K trans , in that order ( P = 0.000 for all, Fig. 5 , Table 4 ).

ROC curve of each variable for independent diagnosis of csPCa

Kendall’s tau-b correlation coefficient showed a significant correlation between PI-RADS score and T2WI score ( t = 0.769, P < 0.001). To avoid multicollinearity among variables, PI-RADS score and T2WI score were separately introduced into the biparametric and multiparametric models.

## Multivariable logistic regression analysis of Semi-quantitative and quantitative parameters

The biparametric model involving ADC measured by bpMRI plus T2WI scores and PSAD, and the multiparametric model involving ADC measured by mpMRI plus PI-RADS score and PSAD were created by binary logistic regression.

The first biparametric model involving ADC measured by bpMRI plus T2WI score and PSAD is shown below:

\(Logit \left(P\right)= -1.925+0.494\times PSAD+1.006 \times T2WI\,scores-2.434 \times ADC\ ;\) in which, the independent variables, including T2WI score (OR = 2.734; 95% CI, 2.199–3.398), ADC (OR = 0.088; 95% CI, 0.029–0.266), and PSAD (OR = 1.639, 95% CI, 1.223–2.197) were all statistically significant ( P < 0.05 for all).

The model could predict 83.8% of csPCa cases, with a positive predictive value of 83.4% and a negative predictive value of 84.2% (χ 2 = 363.055, P < 0.001).

The second multiparametric model involving ADC measured by mpMRI plus PI-RADS score and PSAD is shown below:

\(Logit \left(P\right)= -2.212+0.441\times PSAD+1.120 \times PI-RADS\, scores-2.350 \times ADC\ ;\) in which, the independent variables, including PI-RADS scores (OR = 3.064; 95% CI, 2.428–3.866), ADC (OR = 0.095; 95% CI, 0.032–0.288), and PSAD (OR = 1.554, 95% CI, 1.170–2.064) were all statistically significant ( P < 0.05 for all).

The model predicted 85.2% of csPCa cases with a positive predictive value of 86.4% and a negative predictive value of 84.1% (χ 2 = 376.368, P < 0.001).

## Diagnostic efficiency of the biparametric and multiparametric models in diagnosing csPCa compared with that of PI-RADS

The Areas under curve (AUC) of the multiparametric model was significantly higher than those of the biparametric model and PI-RADS (Delong test P < 0.05, Fig. 6 ). The multiparametric model showed the highest Youden index, followed by the biparametric model (Table 5 ).

ROC curve of each model and PI-RADS score in the diagnosis of csPCa PIS, the 5-point results of PI-RADS v2.1 assessed by senior physicians; PRE1, the biparametric model involving ADC measured by bpMRI plus T2WI scores and PSAD; PRE2, the multiparametric model involving ADC measured by mpMRI plus PI-RADS scores and PSAD; ROC, receiver operating characteristic; PI-RADS, prostate imaging reporting and data system

Thirty cases were selected to verify the accuracy of the prediction model. The sigmoid function was used for the conversion. The average P values (the probability of csPCa occurrence) were 69.87% and 71.36% in the csPCa group, and 16.22% and 15.31% in the non-csPCa group. The top 10 representative ROI verification results are presented in Table 6 .

In the present study, the PI-RADS v2.1 based on the mpMRI findings showed relatively high accuracy (89.5%), sensitivity (78.9%), and specificity (86.6%) for diagnosis of csPCa, which is consistent with those findings in previous studies [ 11 , 12 ]. According to a meta-analysis [ 12 ], the diagnostic sensitivity and specificity of PI-RADS v2.1 for csPCa were 87% and 74%, respectively. However, a study by Westphalen et al. [ 13 ] suggested relatively low positive predictive value of PI-RADS v2.1 for prostate MRI (49%; 95% CI, 40–58%) due to the strong subjectivity in the process of scoring. The independent diagnostic efficiency of quantitative parameters, like ADC (84.6%) and PSAD (80.2%), are relatively low, but these parameters are more objective and measurable, due to the well-recognized standards and measurement repeatability. Pepe et al. [ 14 ] found that ADC evaluation could support clinicians in decision making of patients with PI-RADS score 3 at risk for csPCa, for increase the ROC from 0.71 to 0.81. Marco [ 15 ] found that PSAD can help detected mpMRI false negative csPCa. And several studies have shown that PI-RADS combined with ADC or PSAD significantly enhances the diagnostic accuracy and positive predictive value of csPCa, thus avoiding unnecessary biopsy [ 16 , 17 , 18 ]. DCE MRI is an established mpMRI sequence for assessing prostate cancer, which highlights hemodynamic changes in cancer lesions and measures quantitative parameters that reflect microvascular perfusion (e.g., K ep , K trans ) [ 19 ]. As a single predictor, the odds ratios (OR) of K ep (0.987, P > 0.05) and K trans (0.794, P > 0.05) in the present study were closer to 1, suggesting that they presented no contribution to the predictive model of csPCa, which is consistent with previous findings [ 20 , 21 ]. We created predictive models based on MR imaging data, quantitative parameters, and clinical indicators, which not only significantly enhance the diagnostic accuracy for csPCa, but also objectively identify the cancer lesion. The results were similar with those from Liying Han [ 22 ]. Its area under curve value of the combined model (0.911) was also higher than those of ADC, PSAD, and PI-RADS v2.0 (0.887, 0.861, and 0.859, respectively).

In recent years, a large number of scholars have proposed that prostate MRI without DCE (bpMRI) may replace PI-RADSv2.1 based on mpMRI as a non-invasive monitoring means for csPCa [ 23 , 24 , 25 ]. Comparing the results of a meta-analyses [ 24 ], the diagnostic sensitivity (87%, 95%CI: 78%-93%) and specificity (72%, 95%CI: 56%-84%) of mpMRI for csPCa were not significantly different from bpMRI (sensitivity: 84%, 95%CI: 80% to 88%, specificity 75%, 95%CI: 68% to 81%). However, in our study, the diagnostic efficiency and positive predictive value of multiparametric model were significantly higher than those of the biparametric model. That’s consistent with PI-RADS v2.1, when bpMRI is performed and DCE data are not obtained, transition zone (TZ) assessment remains unchanged, and the proportion of men with PI-RADS assessment category 3 may increase [ 3 ]. Similarly, Greer et al. [ 26 ] found that DCE-MRI was conducive to enhance the diagnostic efficiency for csPCa, and the abnormal findings on DCE-MRI significantly increased the detection rate of PI-RADS v2.0 in categories 2–5 (A total of 163 patients with 654 lesions were evaluated). Therefore, mpMRI can be recommended to avoid misdiagnosis of csPCa, particularly suitable for prostate cancer risk groups.

Although the diagnostic efficiency of multiparametric model for csPCa was superior to that of the biparametric model, the complex procedure may challenge junior physicians or physicians in low-level hospitals with fewer cases. In the present study, the negative predictive value of the biparametric model was comparable to that of the multiparametric model (84.2% vs. 84.1%). Moreover, considering the risks associated with intravenous injection of contrast agents, and low economic and time cost of dynamic contrast-enhanced MRI, we think biparametric model might be more appropriate for early screening of csPCa.

This was a retrospective cohort study with an expanded sample size. Biopsy was not performed prior to mpMRI. Most of the involved subjects were pathologically diagnosed as prostate diseases by MRI/ultrasound fusion-guided biopsy or pathological examination after radical prostatectomy. This helped improve the reliability of our findings. In addition, this is first study to compare the characteristics and clinical value of the biparametric and multiparametric models involving MRI imaging data, quantitative parameters, and clinical indicators with those of PI-RADS v2.1. We identified the critical role of DCE-MRI in diagnosing csPCa, which can make up for the limitations of PI-RADS v2.1. Collectively, the biparametric and multiparametric models were found to be useful tools for selecting the optimal MRI and for planning the therapeutic strategy. However, its accuracy still requires to be verified by a larger sample from multiple centers. The latest research [ 27 , 28 ] showed that MRI scoring with the Prostate Imaging for Recurrence Reporting assessment based on mpMRI could provide structured, reproducible, and accurate evaluation of local recurrence after definitive therapy for prostate cancer. Meanwhile, Pepe’s study shows PSMA PET/CT demonstrated good accuracy in the diagnosis of csPCa, which was not inferior to mpMRI (77.5% vs. 73.7%) [ 29 ]. These will be the focus of further research.

Some limitations of our study should be acknowledged. First of all, this study focused on the diagnosis of csPCa without taking into consideration the stage of cancer lesions. We chose the index lesion of csPCa because it contains lethal progenitor cells that determine the progression and metastasis of prostate cancer [ 30 , 31 ]. Second, there were no independent studies on the diagnostic efficacy of transitional zone and peripheral zone lesions in this study. Third, this was a single-center retrospective cohort study. Our results should be further validated in a larger, multi-center prospective study.

PI-RADS score was the most accurate independent diagnostic index. The predictive value of bpMRI model for csPCa was slightly lower than that of mpMRI model, but higher than that of PI-RADS score. Created bpMRI and mpMRI models for diagnosing csPCa, can overcome the limitations of PI-RADS v2.1 and facilitate treatment decision-making. BpMRI might be more appropriate for early screening of csPCa, and mpMRI for avoiding missed diagnosis.

## Availability of data and materials

The data and materials used during the current study are available from the corresponding author upon reasonable request.

## Abbreviations

Prostate cancer

Multiparametric magnetic resonance imaging

Prostate imaging reporting and data system

Clinically significant prostate cancer

Dynamic contrast-enhanced

Biparametric magnetic resonance imaging

Prostate specific antigen density

Transrectal Ultrasound

MRI-US fusion targeted biopsy

Radical prostatectomy

T2 weighted image

Apparent diffusion coefficient

Volume transfer constant between blood plasma and the extracellular extravascular space

Rate constant between the extracellular extravascular space and the blood plasma

Gleason score

Receiver operating characteristic

Clinically insignificant Pca

International Society of Urological Pathology

Areas under curve

Odds Ratios

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## Acknowledgements

We’d like to thank for the technical supports provided by NHC Key Laboratory of Thalassemia Medicine (Guangxi Medical University).

This work was supported by the Guangxi Zhuang Autonomous Region Health Commission Self-financed Scientific Research Project (Z20201092); and Guangxi Medical University Undergraduate Innovation and Entrepreneurship Training Program (202110598157).

## Author information

Xiao Feng and Xin Chen have contributed equally to this work and share first authorship.

## Authors and Affiliations

Department of Radiology, The First Affiliated Hospital of Guangxi Medical University, No.6 Shuangyong Road, Qingxiu District, Nanning, 530021, Guangxi, P.R. China

Xiao Feng, Peng Peng, He Zhou, Yi Hong, Chunxia Zhu, Libing Lu, Siyu Xie, Sijun Zhang & Liling Long

Department of Radiology, Jiangjin Hospital, Chongqing University, No.725, Jiangzhou Avenue, Dingshan Street, Chongqing, 402260, China

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## Contributions

Xiao Feng: Project development, Data Collection, Data analysis, Manuscript writing. Xin Chen: Project development, Data Collection, Data analysis. Peng Peng: Protocol development. He Zhou: Data Collection, Data analysis. Yi Hong: Data Collection, Data analysis. Chunxia Zhu: Data Collection, Data analysis. Libing Lu: Data Collection, Data analysis. Siyu Xie: Data Collection, Data analysis. Sijun Zhang: Data Collection, Data analysis. Liling Long: design and revision of the manuscript. All authors reviewed the manuscript.

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Feng, X., Chen, X., Peng, P. et al. Values of multiparametric and biparametric MRI in diagnosing clinically significant prostate cancer: a multivariate analysis. BMC Urol 24 , 40 (2024). https://doi.org/10.1186/s12894-024-01411-0

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Introduction A non-zero element of an ordered field is infinitesimal if and only if its absolute value is smaller than any element of of the form , for a standard natural number. Ordered fields that have infinitesimal elements are also called non-Archimedean.

That being said, until we rigorously explain the logical formalism of non-standard analysis, we should caution the reader that typical transferrable statements involve quanti ers over numbers and not sets of numbers. For example, the completeness property for R says that \for all sets of num-bers Athat are nonempty and bounded above, sup(A ...

In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic .

Short introduction to Nonstandard Analysis E. E. Rosinger These lecture notes, to be completed in a later version, offer a short and rigorous introduction to Nostandard Analysis, mainly aimed to reach to a presentation of the basics of Loeb integration, and in particular, Loeb measures.

Nonstandard analysis is a branch of mathematical logic which introduces hyperreal numbers to allow for the existence of "genuine infinitesimals," which are numbers that are less than 1/2, 1/3, 1/4, 1/5, ..., but greater than 0. Abraham Robinson developed nonstandard analysis in the 1960s.

A branch of mathematical logic concerned with the application of the theory of non-standard models to investigations in traditional domains of mathematics: mathematical analysis, function theory, the theory of differential equations, probability theory, and others. The basic method of non-standard analysis can roughly be described as follows.

Nonstandard analysis is a rich formalization of analysis that uses a certain explicit notions of infinitesimal objects. In fact, not only infinitesimal but also infinitely large can be accomodated (and must be).

Nonstandard Analysis Home Textbook Authors: Martin Väth Introductory text focussing on the more advanced applications of nonstandard analysis Preliminary knowledge in model theory not required Deep models for nonstandard embeddings are described with detailed proofs Many exercises, most of them with solutions — 7636 Accesses 9 Citations Sections

An element in an ordered field whose absolute value is less than 1/ n for any positive integer n. Non-archimedean : Infinitesimal or infinite values exist in an ordered field. Transfer principle : A rule which transforms assertions about standard sets, mappings, etc., into one about internal sets, mappings.

1 More than thirty years after its discovery by Abraham Robinson , the ideas and techniques of Nonstandard Analysis (NSA) are being applied across the whole mathematical spectrum,as well as constituting an im portant field of research in their own right. The current methods of NSA now greatly extend Robinson's original work with infinitesimals.

Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then.

Nonstandard Analysis and its Applications Search within full text Get access Cited by 43 Nigel Cutland, University of York Publisher: Cambridge University Press Online publication date: June 2012 Print publication year: 1988 Online ISBN: 9781139172110 DOI: https://doi.org/10.1017/CBO9781139172110 Subjects:

Abstract. In this paper we give an introduction to nonstandard analysis, starting with an ultrapower construction of the hyperreals. We then demon-strate how theorems in standard analysis \transfer over" to nonstandard anal-ysis, and how theorems in standard analysis can be proven using theorems in nonstandard analysis. 1. Introduction

163 As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides some formalism to this type of calculus. So, do you guys think it is a subject worth learning?

Nonstandard Analysis. This mathematical theory has restored infinitesimals to good standing. They had been employed since antiquity, but often with doubts, to solve such problems as finding a ...

Non-standard analysis is a product of logic, hence we will start giving a short intro-duction to logic. We will start with some very basic de nitions. One of the main goals of logic is giving an unambiguous language in which we can talk about math-ematics. This so-called logical language will always contain the following logical

from what in nonstandard analysis is known as the 'transfer' theorem). Often the nonstandard proof is intuitively more attractive, simpler and shorter, which is one of the reasons to be interested in nonstandard analysis at all. Another reason is that totally new mathematical models for all kinds of problems can be (and in

An Introduction to Non Standard Analysis and Applications to Quantum Theory. S. Albeverio. Physics, Mathematics. 1987. We briefly sketch the basic theory of non-standard analysis. We also discuss some applications to quantum theory and related areas (stochastic processes, partial differential equations). 2.

These criticisms are analyzed below. Introduction The evaluation of nonstandard analysis in the literature has varied greatly. Paul Halmos described it as a technical special development in mathematical logic. Terence Tao summed up the advantage of the hyperreal framework by noting that it

Non-standard analysis is an area of mathematics that provides a natural framework for the discussion of infinite economies. It is more suitable in many ways than Lebesgue measure theory as a source of models for large but finite economies since the sets of traders in such models are infinite sets which can be manipulated as though they were finite sets.

By Shaun Regenbaum. This video intuitively explains infinitesimals and the basics of Non-Standard Analysis. There are simplifications of advanced mathematics, just beware. Thanks to DM Ashura (Bill Shillito) for his awesome music and his constant support for my love of math and science. He is truly an inspiration.

More generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. A field that satisfies the transfer principle for real numbers is a hyperreal field, and non-standard real analysis uses these fields as non-standard models of the real numbers. Robinson's original approach was based ...

An Analysis of Living Standards For a […] In this article, we will look into the 30 countries with highest standard of living ranked by GDP (PPP) per capita. If you want to skip our detailed ...

As part of its digital strategy, the EU wants to regulate artificial intelligence (AI) to ensure better conditions for the development and use of this innovative technology. AI can create many benefits, such as better healthcare; safer and cleaner transport; more efficient manufacturing; and cheaper and more sustainable energy.. In April 2021, the European Commission proposed the first EU ...

In due course these structures became known as non-standard models of arithmetic. For nearly thirty years since the appearance of Skolem's paper non-standard models were not used or considered in any sense by the working mathematician. Robinson's fundamental paper, which appeared in 1961 under the title 'Non-standard Analysis', (see [11 ...

1. A similar article was published in Sterne Kessler's PTAB Year in Review 2021. This article has been updated to include new statistics and developments.

Image analysis. Prostate volume was measured according to the PI-RADS v2.1 [] standard for the calculation of PSAD.Prostate index lesions of each patient were scored on a five-point scale on T2WI images (T2WI score) and mpMRI images (PI-RADS score) by a senior radiologist blinded to the pathological results and having 12 years of MRI experience (having read more than eighty thousand patients ...