The uniform continuity theorem states that every pointwise continuous real‐valued function on the unit interval is uniformly continuous. In constructive mathematics, is strictly stronger than the decidable fan theorem, but Loeb [17] has shown that the two principles become equivalent by encoding continuous real‐valued functions as type‐one functions. However, the precise relation between such type‐one functions and continuous real‐valued functions (usually described as type‐two objects) has been unknown. In this paper, we introduce an appropriate notion of continuity for a modulus (...) of a continuous real‐valued function on [0, 1], and show that real‐valued functions with continuous moduli are exactly those functions induced by Loeb's codes. Our characterisation relies on two assumptions: (1) real numbers are represented by regular sequences (equivalently Cauchy sequences with explicitly given moduli); (2) the continuity of a modulus is defined with respect to the product metric on the regular sequences inherited from the Baire space. Our result implies that is equivalent to the statement that every pointwise continuous real‐valued function on [0, 1] with a continuous modulus is uniformly continuous. We also show that is equivalent to a similar principle for real‐valued functions on the Cantor space. These results extend Berger's [2] characterisation of for integer‐valued functions on and unify some characterisations of in terms of functions having continuous moduli. (shrink)

Many systems of thoughts have migrated to Japan from abroad - Buddhism, Taoism, Confucianism, natural science, Marxism; all were transmitted to Japan and stopped there. It is said that Japan has been the terminus for imported thoughts. However, during the process of taking root in Japan almost every idea has been transformed within the Japanese mind. I would like to discuss a typical example of how Christianity has been changed within the Japanese mind.

Ecological developmental biology (eco‐devo) explores the mechanistic relationships between the processes of individual development and environmental factors. Recent studies imply that some of these relationships have deep evolutionary origins, and may even pre‐date the divergences of the simplest extant animals, including cnidarians and sponges. Development of these early diverging metazoans is often sensitive to environmental factors, and these interactions occur in the context of conserved signaling pathways and mechanisms of tissue homeostasis whose detailed molecular logic remain elusive. Efficient methods for (...) transgenesis in cnidarians together with the ease of experimental manipulation in cnidarians and sponges make them ideal models for understanding causal relationships between environmental factors and developmental mechanisms. Here, we identify major questions at the interface between animal evolution and development and outline a road map for research aimed at identifying the mechanisms that link environmental factors to developmental mechanisms in early diverging metazoans.Also watch the Video Abstract. (shrink)

We characterize the category of Sambin’s positive topologies as the result of the Grothendieck construction applied to a doctrine over the category Loc of locales. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the constructed adjunction.

We identify bar recursion on moduli of continuity as a fundamental notion of constructive mathematics. We show that continuous functions from the Baire space \ to the natural numbers \ which have moduli of continuity with bar recursors are exactly those functions induced by Brouwer operations. The connection between Brouwer operations and bar induction allows us to formulate several continuity principles on the Baire space stated in terms of bar recursion on continuous moduli which naturally characterise some variants of bar (...) induction. These principles state that a certain kind of continuous function from \ to \ admits a modulus of continuity with a bar recursor. The results for the Baire space are recast in the setting of the Cantor space \ using the notion of fan recursor, a bar recursor for binary trees. This yields characterisations of uniformly continuous functions on the Cantor space and fan theorem in terms of fan recursors. The results for the Cantor space hold over the extensional version of intuitionistic arithmetic in all finite types ), and those for the Baire space hold over \ extended with the type for Brouwer operations. Our work places Spector’s bar recursion in a proper context of Brouwer’s intuitionistic mathematics and clarifies the connection between bar recursion and bar induction. (shrink)

We present statements equivalent to some fragments of the principle of non-deterministic inductive definitions (NID) by van den Berg (2013), working in a weak subsystem of constructive set theory CZF. We show that several statements in constructive topology which were initially proved using NID are equivalent to the elementary and finitary NIDs. We also show that the finitary NID is equivalent to its binary fragment and that the elementary NID is equivalent to a variant of NID based on the notion (...) of biclosed subset. Our result suggests that proving these statements in constructive topology requires genuine extensions of CZF with the elementary or finitary NID. (shrink)

The strong continuity principle reads “every pointwise continuous function from a complete separable metric space to a metric space is uniformly continuous near each compact image.” We show that this principle is equivalent to the fan theorem for monotone \ bars. We work in the context of constructive reverse mathematics.

Transvertebral magnetic stimulation (TVMS) of the human lumbar spinal cord can evoke bilateral rhythmic leg movements, as in walking, supposedly through the activation of spinal locomotor neural circuitry. However, an appropriate stimulus intensity that can effectively drive the human spinal locomotor circuitry to evoke walking-like movements has not been determined. To address this issue, TVMS was delivered over an intervertebral space of the lumbar cord (L1–L3) at different stimulus intensities (10–70% of maximum stimulator output) in healthy human adults. In a (...) stimulus intensity-dependent manner, TVMS evoked two major patterns of rhythmic leg movements in which the left-right movement cycles were coordinated with different phase relationships: hopping-like movements, in which both legs moved in the same direction in phase, and walking-like movements, in which both legs moved alternatively in anti-phase; uncategorized movements were also observed which could not be categorized as either movement type. Even at the same stimulation site, the stimulus-evoked rhythmic movements changed from hopping-like movements to walking-like movements as stimulus intensity was increased. Different leg muscle activation patterns were engaged in the induction of the hopping- and walking-like movements. The magnitude of the evoked hopping- and walking-like movements was positively correlated with stimulus intensity. The human spinal neural circuitry required a higher intensity of magnetic stimulation to produce walking-like leg movements than to produce hopping-like movements. These results suggest that TVMS activates distinct neural modules in the human spinal cord to generate hopping- and walking-like movements. (shrink)

Studies with primates in sequence production tasks reveal that chimpanzees make action plans before initiating responses and making on-line adjustments to spatially exchanged stimuli, whereas such planning isn't evident in monkeys. Although planning may rely on phylogenetically newer regions in the inferior parietal lobe – along with the frontal lobes and basal ganglia – it dates back to as far as five million years ago.

In the context of constructive reverse mathematics, we show that weak Kőnig's lemma () implies that every pointwise continuous function is induced by a code in the sense of reverse mathematics. This, combined with the fact that implies the Fan theorem, shows that implies the uniform continuity theorem: every pointwise continuous function has a modulus of uniform continuity. Our results are obtained in Heyting arithmetic in all finite types with quantifier‐free axiom of choice.

Following Dirac's generalized canonical formalism, we develop a quantization scheme for theN-dimensional system described by the Lagrangian $L_0 (\dot y,y) = \frac{1}{2}h_{ij} (y)\dot y^i \dot y^j + b_i (y)\dot y^i - w(y)$ which is supposed to be invariant under the gauge transformation $y^i \to y\prime ^i = y^i + (\rho ^i _\alpha + \sigma ^i _{\alpha j} \dot y^j )\delta \Lambda ^\alpha + \tau ^i _\alpha \delta \dot \Lambda ^\alpha$ . The gauge invariance necessarily implies that the Lagrangian is singular. (...) The identities imposed by the gauge invariance are enumerated and reduced to simpler forms. There are primary and secondary constraints, both of which are of first class. The reduced identities are solved explicitly for the case where the secondary constraints constitute the generators of the groupSO(M), and thus an explicit expression for the manifestly gauge-invariant Lagrangian is obtained. By fixing the gauge appropriately, the unphysical variables are eliminated and a quantization is achieved using only physical variables. Our formulation is covariant under an arbitrary point transformation of physical variables. The problem of formulating a quantum action principle is also commented on briefly. (shrink)

A function from Baire space to the natural numbers is called formally continuous if it is induced by a morphism between the corresponding formal spaces. We compare formal continuity to two other notions of continuity on Baire space working in Bishop constructive mathematics: one is a function induced by a Brouwer‐operation (i.e., inductively defined neighbourhood function); the other is a function uniformly continuous near every compact image. We show that formal continuity is equivalent to the former while it is strictly (...) stronger than the latter. The equivalence of formally continuous functions and those induced by Brouwer‐operations requires Countable Choice. (shrink)

ABSTRACTThreat detection is crucial to survival. Studies using unnatural visual scene settings have shown that humans and primates are able to identify snakes more quickl...

dai 1-kan. Sandō heikō hen -- dai 3-kan. Shintō soku kokutai mon -- dai 4-kan. Judō soku keisei mon -- dai 5-6-kan. Butsudō soku gedatsu mon -- dai 7-kan. Shoshi soku hyakka mon -- dai 8-9-kan. Keisei jiron hen -- dai 10-kan. Shisō oyobi shukan hen.

One of our recent studies has revealed that a numerically trained chimpanzee can memorize a correct sequence of five numbers shown on a monitor. Comparative investigations with humans show very similar patterns of errors in the two species, suggesting humans and chimpanzee share homologous memory processes. Whether or not 5 is a pure capacity limit for the chimpanzee remains an empirical question.

Schwinger's action principle is formulated for the quantum system which corresponds to the classical system described by the LagrangianL c( $\dot x$ , x)=(M/2)gij(x) $\dot x$ i $\dot x$ j−v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider onlyc-number variations of coordinates and time. The Euler-Lagrange equation, the canonical commutation relations, and the canonical equations of motion are derived from this principle in a consistent manner. Further, it is shown that an arbitrary point (...) transformation leaves the forms of the fundamental equations invariant. The judicious choice of the quantal Lagrangian is essential in our formulation. A quantum mechanical analog of Noether's theorem, which relates the invariance of the quantal action with a conservation law, is established. The ambiguities in the quantal Lagrangian are also discussed and it is pointed out that the requirement of invariance is not sufficient to determine uniquely the quantal Lagrangian and the Hamiltonian. (shrink)

文明批評家・土田杏村の初期の論説の他、多岐にわたる主題をとおして、人文科学の黎明を考究した画期的な日本思想史研究。.

西洋近代を経験した東アジア各国はいかに固有の哲学を生み出しそして今どんな〈世界哲学〉の構想が可能なのか。国際共同研究の成果。.