The open-domain Frame Problem is the problem of determining what features of an open task environment need to be updated following an action. Here we prove that the open-domain Frame Problem is equivalent to the Halting Problem and is therefore undecidable. We discuss two other open-domain problems closely related to the Frame Problem, the system identification problem and the symbol-grounding problem, and show that they are similarly undecidable. We then reformulate the Frame Problem as a quantum decision problem, and show (...) that it is undecidable by any finite quantum computer. (shrink)

We prove that NIP valued fields of positive characteristic are henselian, and we begin to generalize the known results on dp-minimal fields to dp-finite fields. On any unstable dp-finite field K, we define a type-definable group of “infinitesimals,” corresponding to a canonical group topology on (K, +). We reduce the classification of positive characteristic dp-finite fields to the construction of non-trivial Aut(K/A)-invariant valuation rings.

We give a survey of results obtained in the model theory of finite and pseudo-finite fields.

In this paper, we study the lower semilattice of NP-subspaces of both the standard polynomial time representation and the tally polynomial time representation of a countably infinite dimensional vector space V∞ over a finite field F. We show that for both the standard and tally representation of V∞, there exists polynomial time subspaces U and W such that U + V is not recursive. We also study the NP analogues of simple and maximal subspaces. We show that the existence (...) of P-simple and NP-maximal subspaces is oracle dependent in both the tally and standard representations of V∞. This contrasts with the case of sets, where the existence of NP-simple sets is oracle dependent but NP-maximal sets do not exist. We also extend many results of Nerode and Remmel concerning the relationship of P bases and NP-subspaces in the tally representation of V∞ to the standard representation of V∞. (shrink)

Soit Π une théorie complète de corps pseudo-finis. L'objet de cet article est de montrer que, dans le langage des anneaux augmenté d'un symbole de prédicat unaire (pour le petit corps), la théorie des paires élémentaires non triviales de modèles de Π admet 2n0 complétions, soit le maximum envisageable. /// Let Π be a complete theorie of pseudo-finite fields. In this article we prove that, in the langage of fields to which we add a unary predicate for (...) a substructure, the theory of non trivial elementary pairs of models of Π has 2 ℵ 0 completions, that is, the maximum that could exist. (shrink)

In spite of the analogies between Q p and F p ((t)) which became evident through the work of Ax and Kochen, an adaptation of the complete recursive axiom system given by them for Q p to the case of F p ((t)) does not render a complete axiom system. We show the independence of elementary properties which express the action of additive polynomials as maps on F p ((t)). We formulate an elementary property expressing this action and show that (...) it holds for all maximal valued fields. We also derive an example of a rather simple immediate valued function field over a henselian defectless ground field which is not a henselian rational function field. This example is of special interest in connection with the open problem of local uniformization in positive characteristic. (shrink)

A field K in a ring language $\mathcal {L}$ is finitely undecidable if $\mbox {Cons}(T)$ is undecidable for every nonempty finite $T \subseteq {\mathtt{Th}}(K; \mathcal {L})$. We extend a construction of Ziegler and (among other results) use a first-order classification of Anscombe and Jahnke to prove every NIP henselian nontrivially valued field is finitely undecidable. We conclude (assuming the NIP Fields Conjecture) that every NIP field is finitely undecidable. This work is drawn from the author’s PhD thesis [48, (...) Chapter 3]. (shrink)

If K is a field of finite Morley rank, then for any parameter set $A \subseteq K^{eq}$ the prime model over A is equal to the model-theoretic algebraic closure of A. A field of finite Morley rank eliminates imaginaries. Simlar results hold for minimal groups of finite Morley rank with infinite acl( $\emptyset$ ).

In Ref. 1 we have considered the finite-dimensional quantum mechanics. There the quantum mechanical space of states wasV=C r. It is known that the second quantization of this space is the space of square-summable functions of finite number of variables(L 2(Rr,dx)) (Segal isomorphism). Creation and annihilation operators were introduced in Ref. 1, and the former coincided with the usual position and momentum operators in the conventional quantum mechanics. In this paper we shall investigate the spectral properties of field (...) operators. We shall show that the isomorphism between the exponential ofV andL 2(Rr,dx) can be understood as the decomposition by generalized eigenvectors of field operators (“Fourier transform”). (shrink)

The mathematical model of physical problems interprets physical phenomena closely. This research work is focused on numerical solution of a nonlinear mathematical model of fractional Maxwell nanofluid with the finite difference element method. Addition of nanoparticles in base fluids such as water, sodium alginate, kerosene oil, and engine oil is observed, and velocity profile and heat transfer energy profile of solutions are investigated. The finite difference method involving the discretization of time and distance parameters is applied for numerical (...) results by using the Caputo time fractional operator. These results are plotted against different physical parameters under the effects of magnetic field. These results depicts that a slight decrease occurs for velocity for a high value of Reynolds number, while a small value of Re provides more dominant effects on velocity and temperature profile. It is observed that fractional parameters α and β show inverse behavior against u y, t and θ y, t. An increase in volumetric fraction of nanoparticles in base fluids decreases the temperature profile of fractional Maxwell nanofluids. Using mathematical software of MAPLE, codes are developed and executed to obtain these results. (shrink)

Let G be a finite group. We explore the model-theoretic properties of the class of differential fields of characteristic zero in m commuting derivations equipped with a G-action by differential fie...

We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.

We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a group scheme action.

We give a definition, in the ring language, of Zp inside Qp and of Fp[[t]] inside Fp), which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist (...) a uniform definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Zp inside Qp uniformly for all p. For any fixed finite extension of Qp, we give an existential formula and a universal formula in the ring language which define the valuation ring. (shrink)

A certain class of geometric objects is considered against the background of a classical gauge field associated with an arbitrary structural Lie group. It is assumed that the components of these objects depend on the gauge potentials and their first derivatives, and also on certain gauge-dependent parameters whose properties are suggested by the interaction of an isotopic spin particle with a classical Yang-Mills field. It is shown that the necessary and sufficient conditions for the invariance of the given objects under (...) a finite gauge transformation are embodied in a set of three relations involving the derivatives of their components. As a special case these so-called invariance identities indicate that there cannot exist a gauge-invariant Lagrangian that depends on the gauge potentials, the interaction parameters, and the4-velocity components of a test particle. However, the requirement that the equations of motion that result from such a Lagrangian be gauge-invariant, uniquely determines the structure of these equations. (shrink)

Let K be a countable field. Then a weak presentation of K is an isomorphism of K onto a field whose elements are natural numbers, such that all the field operations are extendible to total recursive functions. Given a pair of two non-finitely generated countable fields contained in some overfield, we investigate under what circumstances the overfield has a weak presentation under which the given fields have images of arbitrary Turing degrees or, in other words, we investigate Turing (...) separability of various pairs of non-finitely generated fields. (shrink)

Exact solutions of epidemic models are critical for identifying the severity and mitigation possibility for epidemics. However, solving complex models can be difficult when interfering conditions from the real-world are incorporated into the models. In this paper, we focus on the generally unsolvable adaptive susceptible-infected-susceptible epidemic model, a typical example of a class of epidemic models that characterize the complex interplays between the virus spread and network structural evolution. We propose two methods based on mean-field approximation, i.e., the first-order mean-field (...) approximation and higher-order mean-field approximation, to derive the exact solutions to ASIS models. Both methods demonstrate the capability of accurately approximating the metastable-state statistics of the model, such as the infection fraction and network density, with low computational cost. These methods are potentially powerful tools in understanding, mitigating, and controlling disease outbreaks and infodemics. (shrink)

In the paper it will be shown that Reichenbach’s Weak Common Cause Principle is not valid in algebraic quantum field theory with locally finite degrees of freedom in general. Namely, for any pair of projections A, B supported in spacelike separated double cones ${\mathcal{O}}_{a}$ and ${\mathcal{O}}_{b}$ , respectively, a correlating state can be given for which there is no nontrivial common cause (system) located in the union of the backward light cones of ${\mathcal{O}}_{a}$ and ${\mathcal{O}}_{b}$ and commuting with the (...) both A and B. Since noncommuting common cause solutions are presented in these states the abandonment of commutativity can modulate this result: noncommutative Common Cause Principles might survive in these models. (shrink)

In this paper we set out the basic model theory of differential fields of characteristic 0, which have finitely many commuting derivations. We give axioms for the theory of differentially closed differential fields with m derivations and show that this theory is ω-stable, model complete, and quantifier-eliminable, and that it admits elimination of imaginaries. We give a characterization of forking and compute the rank of this theory to be ω m + 1.

We prove, by a probabilistic argument, that a class of ω-categorical structures, on which algebraic closure defines a pregeometry, has the finite submodel property. This class includes any expansion of a pure set or of a vector space, projective space or affine space over a finite field such that the new relations are sufficiently independent of each other and over the original structure. In particular, the random graph belongs to this class, since it is a sufficiently independent expansion (...) of an infinite set, with no structure. The class also contains structures for which the pregeometry given by algebraic closure is non-trivial. (shrink)

The present study addresses the fluid transport behaviour of the flow of gold -copper /biomagnetic blood hybrid nanofluid in an inclined irregular stenosis artery as a consequence of varying viscosity and Lorentz force. The nonlinear flow equations are transformed into dimensionless form by using nonsimilar variables. The finite-difference technique is involved in computing the nonlinear transport dimensionless equations. The significant parameters like angle parameter, the Hartmann number, changing viscosity, constant heat source, the Reynolds number, and nanoparticle volume fraction on (...) the flow field are exhibited through figures. Present results disclose that the Lorentz force strongly lessens the hybrid nanofluid velocity. Elevating the Grashof number values enhances the rate of blood flow. Growing values of the angle parameter cause to reduce the resistance impedance on the wall. Hybrid nanoparticles have a superior wall shear stress than copper nanoparticles. The heat transfer rate is amplifying at the axial direction with the growing values of nanoparticles concentration. The applied Lorentz force significantly reduces the hybrid and unitary nanofluid flow rate in the axial direction. The hybrid nanoparticles expose a supreme rate of heat transfer than the copper nanoparticles in a blood base fluid. Compared to hybrid and copper nanofluid, the blood base fluid has a lower temperature. (shrink)

Suppose T is a weakly minimal theory and p a strong 1-type having locally finite but nontrivial geometry. That is, for any M [boxvR] T and finite Fp, there is a finite Gp such that acl∩p = gεGacl∩pM; however, we cannot always choose G = F. Then there are formulas θ and E so that θεp and for any M[boxvR]T, E defines an equivalence relation with finite classes on θ/E definably inherits the structure of either a (...) projective or affine space over some finite field. We then specify what other structure θ/E may inherit: there is some collection of definable subspaces of finite codimension and some set of algebraic points, which in the affine case may be in the canonically associated vector space, Up to acl, no further structure is possible. If we assume T is weakly minimal and has a strong type p as above, and also that T is unidimensional, we obtain a global description of any model of T in terms of those structures mentioned in the previous paragraph. (shrink)

In this paper we consider classes of finite structures where we have good control over the sizes of the definable sets. The motivating example is the class of finite fields: it was shown in [1] that for any formulain the language of rings, there are finitely many pairs (d,μ) ∈ω×Q>0so that in any finite fieldFand for any ā ∈Fmthe size |ø(Fn,ā)| is “approximately”μ|F|d. Essentially this is a generalisation of the classical Lang-Weil estimates from the category of (...) varieties to that of the first-order-definable sets. (shrink)

Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group. In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser (...) is not injective. (shrink)

This article introduces the notion of duplex elements of the finite rings and corresponding neutrosophic rings. The authors establish duplex ring Dup(R) and neutrosophic duplex ring Dup(R)I)) by way of various illustrations. The tables of different duplicities are constructed to reveal the comparison between rings Dup(Zn), Dup(Dup(Zn)) and Dup(Dup(Dup(Zn ))) for the cyclic ring Zn . The proposed duplicity structures have several algebraic systems with dissimilar consequences. Author’s characterize finite rings with R + R is different from the (...) duplex ring Dup(R). However, this characterization supports that R + R = Dup(R) for some well known rings, namely zero rings and finite fields. (shrink)

Universality has been an important concept in computable structure theory. A class C of structures is universal if, informally, for any structure of any kind there is a structure in C with the same computability-theoretic properties as the given structure. Many classes such as graphs, groups, and fields are known to be universal. This paper is about the class of finitely generated groups. Because finitely generated structures are relatively simple, the class of finitely generated groups has no hope of (...) being universal. We show that finitely generated groups are as universal as possible, given that they are finitely generated: for every finitely generated structure, there is a finitely generated group which has the same computability-theoretic properties. The same is not true for finitely generated fields. We apply the results of this investigation to quasi Scott sentences, and also answer a question of Alvir, Knight, and McCoy. (shrink)

In [4] the authors studied the residue field of a model M of IΔ0 + Ω1 for the principal ideal generated by a prime p. One of the main results is that M/ has a unique extension of each finite degree. In this paper we are interested in understanding the structure of any quotient field of M, i.e. we will study the quotient M/I for I a maximal ideal of M. We prove that any quotient field of M satisfies (...) the property of having a unique extension of each finite degree. We will use some of Cherlin's ideas from [3], where he studies the ideal theory of non standard algebraic integers. (shrink)

The grounding of semiotics in the finiteness of cognition is extended into constructs and methods for analysis by incorporating the assumption that cognition can be similar within and between agents. After examining and formalizing cognitive similarity as an ontological commitment, the recurrence of cognitive states is examined in terms of a “cognitive set.” In the individual, the cognitive set is seen as evolving under the bidirectional, cyclical determination of thought by the historical environment. At the population level, the distributed “global” (...) cognitive set is argued to be constrained to a manifold in which the cognition of individuals is determined only when their cognitive sets meet certain conditions in the world: a result seen as consistent with Lotman’s semiosphere. With these foundations in place, dimensional modelling of the semiosic field is inaugurated. Firstly, measures of cognitive similarity are formalized as cognitive “distance” and on this basis the concept of a semiotic vector is defined. Secondly, semiotic vectors are seen to shape a general pattern of oscillation in semiosis, and thus to imply zero points in semiosic potential. Thirdly, semiosic oscillation in individual agents is shown to be consistent with a novel diachronic or longitudinal interpretation of Greimas’ semiotic square expanded into a “semiotic pipe” in which cognition traverses an n-dimensional space structured by axes of oscillation. Finally, the expanded theory of finite semiotics is advanced as a useful basis for two new complementary disciplines: a computational, mathematical science of “natural semiotic processing” to trace and model semiotic vectors and oscillation; and an ethical, rhetorical art of “technological influencing” to guide its inputs and applications. (shrink)

We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class K. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class K of algebraic systems. One characterization of this concept states that A is locally embedded in K iff it is a subsystem of an ultraproduct of systems from K. In this paper we obtain a similar characterization (...) of approximability of a locally compact system A by systems from K using the language of nonstandard analysis. In the signature of A we introduce positive bounded formulas and their approximations; these are similar to those introduced by Henson [14] for Banach space structures (see also [15, 16]). We prove that a positive bounded formula φ holds in A if and only if all precise enough approximations of φ hold in all precise enough approximations of A. We also prove that a locally compact field cannot be approximated arbitrarily closely by finite (associative) rings (even if the rings are allowed to be non-commutative). Finite approximations of the field R can be considered as possible computer systems for real arithmetic. Thus, our results show that there do not exist arbitrarily accurate computer arithmetics for the reals that are associative rings. (shrink)

When [Formula: see text] is a finite field, [J. Becker, J. Denef and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, in Model Theory of Algebra and Arithmetic, Proceedings Karpacz, Poland, Lecture Notes in Mathematics, Vol. 834 (Springer, Berlin, 1979)] observed that the total residue map [Formula: see text], which picks out the constant term of the Laurent series, is definable in the language of rings with a parameter for [Formula: see text]. Driven by (...) this observation, we study the theory [Formula: see text] of valued fields equipped with a linear form [Formula: see text] which restricts to the residue map on the valuation ring. We prove that [Formula: see text] does not admit a model companion. In addition, we show that [Formula: see text] is undecidable whenever [Formula: see text] is an infinite field. As a consequence, we get that [Formula: see text] is undecidable, where [Formula: see text] maps [Formula: see text] to its complex residue at [Formula: see text]. (shrink)

We give an example of two ordered structures $\mathcal {M},\mathcal {N}$ in the same language $\mathcal {L}$ with the same universe, the same order and admitting the same one-variable definable subsets such that $\mathcal {M}$ is a model of the common theory of o-minimal $\mathcal {L}$ -structures and $\mathcal {N}$ admits a definable, closed, bounded, and discrete subset and a definable injective self-mapping of that subset which is not surjective. This answers negatively two question by Schoutens; the first being whether (...) there is an axiomatization of the common theory of o-minimal structures in a given language by conditions on one-variable definable sets alone. The second being whether definable completeness and type completeness imply the pigeonhole principle. It also partially answers a question by Fornasiero asking whether definable completeness of an expansion of a real closed field implies the pigeonhole principle. (shrink)

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.