We show that any formula with two free variables in a Vapnik–Chervonenkis (VC) minimal theory has VC-codensity at most 2. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where acleq= dcleq, the VC-codensity of a formula is at most the number of free variables (from the work of Aschenbrenner et al., the author, and Laskowski).

We show that in the theory of infinite trees the VC-function is optimal. This generalizes a result of Simon showing that trees are dp-minimal.

Given a model-complete theory of topological fields, we considered its generic differential expansions and under a certain hypothesis of largeness, we axiomatised the class of existentially closed ones. Here we show that a density result for definable types over definably closed subsets in such differential topological fields. Then we show two transfer results, one on the VC-density and the other one, on the combinatorial property NTP2.

This paper presents some finite combinatorics of set systems with applications to model theory, particularly the study of dependent theories. There are two main results. First, we give a way of producing lower bounds on $\mathrm {VC}_{\mathrm {ind}}$-density and use it to compute the exact $\mathrm {VC}_{\mathrm {ind}}$-density of polynomial inequalities and a variety of geometric set families. The main technical tool used is the notion of a maximum set system, which we juxtapose to indiscernibles. In the second (...) part of the paper we give a maximum set system analogue to Shelah’s characterization of stability using indiscernible sequences. (shrink)

In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablity and show that this lies strictly between VC-minimality and dp-minimality. To do this we prove a general result about set systems with independence dimension ≤ 1. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, (...) show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density. (shrink)

We define a new type of “shatter function” for set systems that satisfies a Sauer–Shelah type dichotomy, but whose polynomial-growth case is governed by Shelah’s two-rank instead of VC dimension. We identify the least exponent bounding the rate of growth of the shatter function, the quantity analogous to VC density, with Shelah’s $\omega $ -rank.

We study the Vapnik–Chervonenkis density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite $\mathrm {U}$-rank theories without the finite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of definable families.

In this paper, we study Vapnik‐Chervonenkis density (VC‐density) over indiscernible sequences (denoted VCind‐density). We answer an open question in [1], showing that VCind‐density is always integer valued. We also show that VCind‐density and dp‐rank coincide in the natural way.

Let $\mathcal {S}$ be a family of nonempty sets with VC-codensity less than $2$. We prove that, if $\mathcal {S}$ has the $(\omega,2)$ -property (for any infinitely many sets in $\mathcal {S}$, at least two among them intersect), then $\mathcal {S}$ can be partitioned into finitely many subfamilies, each with the finite intersection property. If $\mathcal {S}$ is definable in some first-order structure, then these subfamilies can be chosen definable too.This is a strengthening of the case $q=2$ of the definable (...) $(p,q)$ -conjecture in model theory [9] and the Alon–Kleitman–Matoušek $(p,q)$ -theorem in combinatorics [6]. (shrink)

A theory $T$ is shown to have an ICT pattern of depth $k$ in $n$ variables iff it interprets some $k$ -maximum VC class in $n$ parameters.

That humans can categorize in different ways does not imply that there are qualitatively distinct underlying natural kinds or that the field of concepts splinters. Rather, it implies that the unitary goal of forming concepts is important enough that it receives redundant expression in cognition. Categorization science focuses on commonalities involved in concept learning. Eliminating makes this more difficult.

It has consistently been shown that agents judge the intervals between their actions and outcomes as compressed in time, an effect named intentional binding. In the present work, we investigated whether this effect is result of prior bias volunteers have about the timing of the consequences of their actions, or if it is due to learning that occurs during the experimental session. Volunteers made temporal estimates of the interval between their action and target onset , or between two events . (...) Our results show that temporal estimates become shorter throughout each experimental block in both conditions. Moreover, we found that observers judged intervals between action and outcomes as shorter even in very early trials of each block. To quantify the decrease of temporal judgments in experimental blocks, exponential functions were fitted to participants’ temporal judgments. The fitted parameters suggest that observers had different prior biases as to intervals between events in which action was involved. These findings suggest that prior bias might play a more important role in this effect than calibration-type learning processes. (shrink)

If the density matrix is treated as an objective description of individual systems, it may become possible to attribute the same objective significance to statistical mechanical properties, such as entropy or temperature, as to properties such as mass or energy. It is shown that the de Broglie--Bohm interpretation of quantum theory can be consistently applied to density matrices as a description of individual systems. The resultant trajectories are examined for the case of the delayed choice interferometer, for which (...) Bell [Int. J. Quantum chem. 155--159, (1980)] appears to suggest that such an interpretation is not possible. Bell’s argument is shown to be based upon a different understanding of the density matrix to that proposed here. (shrink)

The following strong form of density of definable types is introduced for theoriesTadmitting a fibered dimension functiond: given a modelMofTand a definable setX⊆Mn, there is a definable typepinX, definable over a code forXand of the samed-dimension asX. Both o-minimal theories and the theory of closed ordered differential fields are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.

Realism about quantum theory naturally leads to realism about the quantum state of the universe. It leaves open whether it is a pure state represented by a wave function, or an impure (mixed) one represented by a density matrix. I characterize and elaborate on Density Matrix Realism, the thesis that the universal quantum state is objective but can be impure. To clarify the thesis, I compare it with Wave Function Realism, explain the conditions under which they are empirically (...) equivalent, consider two generalizations of Density Matrix Realism, and answer some frequently asked questions. I end by highlighting an implication for scientific realism. (shrink)

It is well known that density matrices can be used in quantum mechanics to represent the information available to an observer about either a system with a random wave function (“statistical mixture”) or a system that is entangled with another system (“reduced density matrix”). We point out another role, previously unnoticed in the literature, that a density matrix can play: it can be the “conditional density matrix,” conditional on the configuration of the environment. A precise definition (...) can be given in the context of Bohmian mechanics, whereas orthodox quantum mechanics is too vague to allow a sharp definition, except perhaps in special cases. In contrast to statistical and reduced density matrices, forming the conditional density matrix involves no averaging. In Bohmian mechanics with spin, the conditional density matrix replaces the notion of conditional wave function, as the object with the same dynamical significance as the wave function of a Bohmian system. (shrink)

This paper is an attempt to count the proportion of tautologies of some intermediate logics among all formulas. Our interest concentrates especially on Medvedev’s logic and its fragment over language with one propositional variable.

In this paper we prove that the preordering $\lesssim $ of provable implication over any recursively enumerable theory $T$ containing a modicum of arithmetic is uniformly dense. This means that we can find a recursive extensional density function $F$ for $\lesssim $. A recursive function $F$ is a density function if it computes, for $A$ and $B$ with $A\lnsim B$, an element $C$ such that $A\lnsim C\lnsim B$. The function is extensional if it preserves $T$-provable equivalence. Secondly, we (...) prove a general result that implies that, for extensions of elementary arithmetic, the ordering $\lesssim $ restricted to $\Sigma_{n}$-sentences is uniformly dense. In the last section we provide historical notes and background material. (shrink)

In noun phrase coordinate constructions, there is a strong tendency for the syntactic structure of the second conjunct to match that of the first; the second conjunct in such constructions is therefore low in syntactic information. The theory of uniform information density predicts that low-information syntactic constructions will be counterbalanced by high information in other aspects of that part of the sentence, and high-information constructions will be counterbalanced by other low-information components. Three predictions follow: lexical probabilities will be lower (...) in second conjuncts than first conjuncts; lexical probabilities will be lower in matching second conjuncts than nonmatching ones; and syntactic repetition should be especially common for low-frequency NP expansions. Corpus analysis provides support for all three of these predictions. (shrink)

We present a population density and moment-based description of the stochastic dynamics of domain $${\text{Ca}}^{2+}$$ -mediated inactivation of L-type $${\text{Ca}}^{2+}$$ channels. Our approach accounts for the effect of heterogeneity of local $${\text{Ca}}^{2+}$$ signals on whole cell $${\text{Ca}}^{2+}$$ currents; however, in contrast with prior work, e.g., Sherman et al. :985–995, 1990), we do not assume that $${\text{Ca}}^{2+}$$ domain formation and collapse are fast compared to channel gating. We demonstrate the population density and moment-based modeling approaches using a 12-state Markov (...) chain model of an L-type $${\text{Ca}}^{2+}$$ channel introduced by Greenstein and Winslow :2918–2945, 2002). Simulated whole cell voltage clamp responses yield an inactivation function for the whole cell $${\text{Ca}}^{2+}$$ current that agrees with the traditional approach when domain dynamics are fast. We analyze the voltage-dependence of $${\text{Ca}}^{2+}$$ inactivation that may occur via slow heterogeneous domain [ $${\text{Ca}}^{2+}$$ ]. Next, we find that when channel permeability is held constant, $${\text{Ca}}^{2+}$$ -mediated inactivation of L-type channels increases as the domain time constant increases, because a slow domain collapse rate leads to increased mean domain [ $${\text{Ca}}^{2+}$$ ] near open channels; conversely, when the maximum domain [ $${\text{Ca}}^{2+}$$ ] is held constant, inactivation decreases as the domain time constant increases. Comparison of simulation results using population densities and moment equations confirms the computational efficiency of the moment-based approach, and enables the validation of two distinct methods of truncating and closing the open system of moment equations. In general, a slow domain time constant requires higher order moment truncation for agreement between moment-based and population density simulations. (shrink)

We prove several theorems about the cardinal $\mathfrak{g}$ associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps fromω toω, all families of size $< \mathfrak{g}$ are below all unbounded families. With respect to a natural ordering of filters onω, all filters generated by $< \mathfrak{g}$ sets are below all non-feeble filters. If $\mathfrak{u}< \mathfrak{g}$ then $\mathfrak{b}< \mathfrak{u}$ and $\mathfrak{g} = \mathfrak{d} = \mathfrak{c}$ . (The definitions of these cardinals are recalled in the introduction.) (...) Finally, some consequences deduced from $\mathfrak{u}< \mathfrak{g}$ by Laflamme are shown to be equivalent to $\mathfrak{u}< \mathfrak{g}$. (shrink)

This paper proposes a formalization of the class of sentences quantified by most, which is also interpreted as proportion of or majority of depending on the domain of discourse. We consider sentences of the form “Most A are B”, where A and B are plural nouns and the interpretations of A and B are infinite subsets of \. There are two widely used semantics for Most A are B: \ > C \) and \ > \dfrac{C}{2} \), where C denotes (...) the cardinality of a given finite set X. Although is more descriptive than, it also produces a considerable amount of insensitivity for certain sets. Since the quantifier most has a solid cardinal behaviour under the interpretation majority and has a slightly more statistical behaviour under the interpretation proportional of, we consider an alternative approach in deciding quantity-related statements regarding infinite sets. For this we introduce a new semantics using natural density for sentences in which interpretations of their nouns are infinite subsets of \, along with a list of the axiomatization of the concept of natural density. In other words, we take the standard definition of the semantics of most but define it as applying to finite approximations of infinite sets computed to the limit. (shrink)

We show that there exist uncountably many pairwise nonisomorphic density-like ideals on $\omega $ which are not generalized density ideals. In addition, they are nonpathological. This answers a question posed by Borodulin-Nadzieja et al. in [this Journal, vol. 80, pp. 1268–1289]. Lastly, we provide sufficient conditions for a density-like ideal to be necessarily a generalized density ideal.

We review the early works which were precursors of the Conceptual Density Functional Theory. Starting from Thomas–Fermi approximation and from the exact formulation of Density Functional Theory by Hohenberg and Kohn’s theorem, we will introduce electronegativity and the theory of hard and soft acids and bases. We will also present a general introduction to the Fukui functions, and their relation with nucleophilicity and electrophilicity, with an emphasis towards the importance of these concepts for chemical reactivity.

We show that for any ω-r.e. degree d and n-r.e. degree b with d

Mathematical Logic in Philosophy of Mathematics

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The main result of this paper is an improvement of the upper bound on the cardinal invariant $cov^*(L_0)$ that was discovered in [11]. Here $L_0$ is the ideal of subsets of the set of natural numbers that have asymptotic density zero. This improved upper bound is also dualized to get a better lower bound on the cardinal $non^*(L_0)$. En route some variations on the splitting number are introduced and several relationships between these variants are proved.

Nelson Goodman’s theory of symbol systems expounded in his Languages of Art has been frequently criticized on many counts (cf. list of secondary literature in the entry “Goodman’s Aesthetics” of Stanford Encyclopedia of Philosophy and Sect. 3 below). Yet it exerts a strong influence and is treated as one of the major twentieth-century theories on the subject. While many of Goodman’s controversial theses are criticized, the technical notions he used to formulate them seem to have been treated as neutral tools. (...) One such technical notion is that of the density of symbol systems. This serves to distinguish linguistic symbols from pictorial representations (after Goodman entirely rejected resemblance in that role) and is a crucial part of Goodman’s explanation of what constitutes aesthetic experience (and so indirectly what is art). Thus its significance for Goodman’s theory is fundamental. The aim of this paper is a detailed, logical analysis of this notion. It turns out that Goodman’s definition is highly problematic and cannot be applied to symbol systems in the way Goodman envisaged. To conclude, Goodman’s theory is problematic not just because of its controversial theses but also because of logical problems with the technical notions used at its very core. Hence the controversial claims are not simply contestable, but inaccurately expressed. (shrink)

The partial ordering of Medvedev reducibility restricted to the family of Π0 1 classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a Π0 1 class, which we call a ``c.e. separating class''. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes.

Concerning Post's problem for Kleene degrees and its relativization, Hrbacek showed in [1] and [2] that if V = L, then Kleene degrees of coanalytic sets are dense, and then for all K ⊆ωω, there are N1 sets which are Kleene semirecursive in K and not Kleene recursive in each other and K. But the density of Kleene semirecursive in K Kleene degrees is not obtained from these theorems. In this note, we extend these theorems by showing that if (...) V = L, then for all K ⊆ ωω, Kleene semirecursive in K Kleene degrees are dense above K. (shrink)

The human brain shows neuroplastic adaptations induced by motor skill training. However, the description of the plastic architecture of the whole-brain network in resting-state is still limited. In the present study, we aimed to detect how motor training affected the density distribution of whole-brain resting-state functional connectivity (FC) brain in fast-ball student-athletes using resting-state functional magnetic resonance imaging (fMRI) data of student-athletes (SA), and non-athlete healthy controls (NC). The voxel-wise data-driven graph theory approach, namely global functional connectivity density (...) (gFCD) mapping was applied. The results showed that the SA group exhibited significantly decreased gFCD in brain regions centered at left triangular part of inferior frontal gyrus (IFG), extending to opercular part of left IFG and middle frontal gyrus (MFG) compared with NC group. The findings supported the idea of an increased neural efficiency of athletes’ brain in the brain regions associated with attentional-motor modulation and executive control. Furthermore, the behavioral results demonstrated that in the SA group, faster executive control reaction time was associated with smaller gFCD values in left IFG. The findings implied that the motor skill training would decrease the numbers of FC in IFG to accelerate the execution control with high attentional demands, and focus the attention to the target detection the athletes are interested in. (shrink)

Analyzing the effective content of the Lebesgue density theorem played a crucial role in some recent developments in algorithmic randomness, namely, the solutions of the ML-covering and ML-cupping problems. Two new classes of reals emerged from this inquiry: thepositive density pointswith respect toeffectively closed sets of reals, and a proper subclass, thedensity-one points. Bienvenu, Hölzl, Miller, and Nies have shown that the Martin-Löf random positive density points are exactly the ones that do not compute the halting problem. (...) Treating this theorem as our starting point, we present several new results that shed light on how density, randomness, and computational strength interact. (shrink)

The Sacks Density Theorem [7] states that the Turing degrees of the recursively enumerable sets are dense. We show that the Density Theorem holds in every model of P - + BΣ 2 . The proof has two components: a lemma that in any model of P - + BΣ 2 , if B is recursively enumerable and incomplete then IΣ 1 holds relative to B and an adaptation of Shore's [9] blocking technique in α-recursion theory to models (...) of arithmetic. (shrink)

We classify the asymptotic densities of the sets according to their level in the Ershov hierarchy. In particular, it is shown that for, a real is the density of an n‐c.e. set if and only if it is a difference of left‐ reals. Further, we show that the densities of the ω‐c.e. sets coincide with the densities of the sets, and there are ω‐c.e. sets whose density is not the density of an n‐c.e. set for any.

We consider effective versions of two classical theorems, the Lebesgue density theorem and the Denjoy–Young–Saks theorem. For the first, we show that a Martin-Löf random real z ∈ [0, 1] is Turing incomplete if and only if every effectively closed class.

The paper explains in what sense the GRW matter density theory is a primitive ontology theory of quantum mechanics and why, thus conceived, the standard objections against the GRW formalism do not apply to GRWm. We consider the different options for conceiving the quantum state in GRWm and argue that dispositionalism is the most attractive one.

We clarify different definitions of the density matrix by proposing the use of different names, the full density matrix for a single-closed quantum system, the compressed density matrix for the averaged single molecule state from an ensemble of molecules, and the reduced density matrix for a part of an entangled quantum system, respectively. We show that ensembles with the same compressed density matrix can be physically distinguished by observing fluctuations of various observables. This is in (...) contrast to a general belief that ensembles with the same compressed density matrix are identical. Explicit expression for the fluctuation of an observable in a specified ensemble is given. We have discussed the nature of nuclear magnetic resonance quantum computing. We show that the conclusion that there is no quantum entanglement in the current nuclear magnetic resonance quantum computing experiment is based on the unjustified belief that ensembles having the same compressed density matrix are identical physically. Related issues in quantum communication are also discussed. (shrink)

The derivation of the expression for the density matrix of scattered particles in terms of that of the incident ones, taking different impact parameters into account, shows that under well-specified and realistic conditions, the final density matrix is of the same kind as the initial one. Thus the final mixed state after a collision can be used directly as the initial mixed state in a subsequent collision. Contrary to a recent claim by Band and Park, there are no (...) “fundamental difficulties with quantum mechanical collision theory.”. (shrink)

We study the relationship between the cofinality $c(Sym(\omega))$ of the infinite symmetric group and the cardinal invariants $\frak{u}$ and $\frak{g}$ . In particular, we prove the following two results. Theorem 0.1 It is consistent with ZFC that there exists a simple $P_{\omega_{1}}$ -point and that $c(Sym(\omega)) = \omega_{2} = 2^{\omega}$ . Theorem 0.2 If there exist both a simple $P_{\omega_{1}}$ -point and a $P_{\omega_{2}}$ -point, then $c(Sym(\omega)) = \omega_{1}$.

This paper is devoted to Landau's concept of the problem of damping in quantum mechanics. It shows that Landau's density matrix formalism should survive in the context of modern quantum electrodynamics. The correct generalized master equation has been derived for the reduced dynamics of the charges. The recent relativistic theory of spontaneous emission becomes reproducible.