8 research outputs found
Creation of the concept of zero point method in teaching mathematics
Pupils learn different calculating algorithms. The effective use of learned algorithms requires creativity in their application to solve diverse tasks. To achieve this goal, it is necessary to create a concept of the calculating algorithm for pupils. The present paper describes a method of creating a zero point method. Teaching of this method is divided into two stages. In the first stage, the student masters the basic algorithm and becomes familiar with the main ideas of this method, while in the second stage a student learns how to apply this method with some modifications in other types of tasks. In our article we present the application of zero point method in solving quadratic inequalities
Almost Quantum Correlations are Inconsistent with Specker's Principle
Ernst Specker considered a particular feature of quantum theory to be
especially fundamental, namely that pairwise joint measurability of sharp
measurements implies their global joint measurability
(https://vimeo.com/52923835). To date, Specker's principle seemed incapable of
singling out quantum theory from the space of all general probabilistic
theories. In particular, its well-known consequence for experimental
statistics, the principle of consistent exclusivity, does not rule out the set
of correlations known as almost quantum, which is strictly larger than the set
of quantum correlations. Here we show that, contrary to the popular belief,
Specker's principle cannot be satisfied in any theory that yields almost
quantum correlations.Comment: 17 pages + appendix. 5 colour figures. Comments welcom
Monotones in General Resource Theories
A central problem in the study of resource theories is to find functions that
are nonincreasing under resource conversions---termed monotones---in order to
quantify resourcefulness. Various constructions of monotones appear in many
different concrete resource theories. How general are these constructions? What
are the necessary conditions on a resource theory for a given construction to
be applicable? To answer these questions, we introduce a broad scheme for
constructing monotones. It involves finding an order-preserving map from the
preorder of resources of interest to a distinct preorder for which nontrivial
monotones are previously known or can be more easily constructed; these
monotones are then pulled back through the map. In one of the two main classes
we study, the preorder of resources is mapped to a preorder of sets of
resources, where the order relation is set inclusion, such that monotones can
be defined via maximizing or minimizing the value of a function within these
sets. In the other class, the preorder of resources is mapped to a preorder of
tuples of resources, and one pulls back monotones that measure the amount of
distinguishability of the different elements of the tuple (hence its
information content). Monotones based on contractions arise naturally in the
latter class, and, more surprisingly, so do weight and robustness measures. In
addition to capturing many standard monotone constructions, our scheme also
suggests significant generalizations of these. In order to properly capture the
breadth of applicability of our results, we present them within a novel
abstract framework for resource theories in which the notion of composition is
independent of the types of the resources involved (i.e., whether they are
states, channels, combs, etc.).Comment: 39 pages + 16 page appendix. 7 figure
A Framework for Universality in Physics, Computer Science, and Beyond
Turing machines and spin models share a notion of universality according to
which some simulate all others. Is there a theory of universality that captures
this notion? We set up a categorical framework for universality which includes
as instances universal Turing machines, universal spin models, NP completeness,
top of a preorder, denseness of a subset, and more. By identifying necessary
conditions for universality, we show that universal spin models cannot be
finite. We also characterize when universality can be distinguished from a
trivial one and use it to show that universal Turing machines are non-trivial
in this sense. Our framework allows not only to compare universalities within
each instance, but also instances themselves. We leverage a Fixed Point Theorem
inspired by a result of Lawvere to establish that universality and negation
give rise to unreachability (such as uncomputability). As such, this work sets
the basis for a unified approach to universality and invites the study of
further examples within the framework.Comment: 77 pages, 12 figures, many diagram
Dilations and information flow axioms in categorical probability
We study the positivity and causality axioms for Markov categories as
properties of dilations and information flow in Markov categories, and in
variations thereof for arbitrary semicartesian monoidal categories. These help
us show that being a positive Markov category is merely an additional property
of a symmetric monoidal category (rather than extra structure). We also
characterize the positivity of representable Markov categories and prove that
causality implies positivity, but not conversely. Finally, we note that
positivity fails for quasi-Borel spaces and interpret this failure as a privacy
property of probabilistic name generation.Comment: 42 page
De Finetti's Theorem in Categorical Probability
We present a novel proof of de Finetti's Theorem characterizing
permutation-invariant probability measures of infinite sequences of variables,
so-called exchangeable measures. The proof is phrased in the language of Markov
categories, which provide an abstract categorical framework for probability and
information flow. The diagrammatic and abstract nature of the arguments makes
the proof intuitive and easy to follow. We also show how the usual
measure-theoretic version of de Finetti's Theorem for standard Borel spaces is
an instance of this result.Comment: 26 pages. v3: referee's suggestions incorporate
Representable Markov categories and comparison of statistical experiments in categorical probability
Markov categories are a recent categorical approach to the mathematical foundations of probability and statistics. Here, this approach is advanced by stating and proving equivalent conditions for second-order stochastic dominance, a widely used way of comparing probability distributions by their spread. Furthermore, we lay the foundation for the theory of comparing statistical experiments within Markov categories by stating and proving the classical Blackwell–Sherman–Stein Theorem. Our version not only offers new insight into the proof, but its abstract nature also makes the result more general, automatically specializing to the standard Blackwell–Sherman–Stein Theorem in measure-theoretic probability as well as a Bayesian version that involves prior-dependent garbling. Along the way, we define and characterize representable Markov categories, within which one can talk about Markov kernels to or from spaces of distributions. We do so by exploring the relation between Markov categories and Kleisli categories of probability monads
Absolute continuity, supports and idempotent splitting in categorical probability
Markov categories have recently turned out to be a powerful high-level
framework for probability and statistics. They accommodate purely categorical
definitions of notions like conditional probability and almost sure equality,
as well as proofs of fundamental results such as the Hewitt-Savage 0/1 Law, the
de Finetti Theorem and the Ergodic Decomposition Theorem. In this work, we
develop additional relevant notions from probability theory in the setting of
Markov categories. This comprises improved versions of previously introduced
definitions of absolute continuity and supports, as well as a detailed study of
idempotents and idempotent splitting in Markov categories. Our main result on
idempotent splitting is that every idempotent measurable Markov kernel between
standard Borel spaces splits through another standard Borel space, and we
derive this as an instance of a general categorical criterion for idempotent
splitting in Markov categories.Comment: 84 pages (including 18 page appendix and many string diagrams). v2:
Corollary 4.4.10 and results needed to establish it were adde