An elementary property of correlations

International audience We study the "shift-Ramanujan expansion" to obtain a formulae for the shifted convolution sum $C_{f,g} (N,a)$ of general functions f, g satisfying Ramanujan Conjecture; here, the shift-Ramanujan expansion is with respect to a shift factor a > 0. Assuming Delange Hypothesis for the correlation, we get the "Ramanujan exact explicit formula", a kind of finite shift-Ramanujan expansion. A noteworthy case is when f = g = Λ, the von Mangoldt function; so $C_{\Lamda, \Lambda} (N, 2k)$, for natural k, corresponds to 2k-twin primes; under the assumption of Delange Hypothesis, we easily obtain the proof of Hardy-Littlewood Conjecture for this case.

A Combinatorial Proof of a Result on Generalized Lucas Polynomials

We give a combinatorial proof of an elementary property of generalized Lucas polynomials, inspired by [1]. These polynomials in s and t are defined by the recurrence relation 〈n〉 = s〈n-1〉+t〈n-2〉 for n ≥ 2. The initial values are 〈0〉 = 2; 〈1〉= s, respectively.

On Chasles' Property of the Helicoid in Tri-Twisted Real Ambient Space

An elementary property of the helicoid is that at every point of the surface the following condition holds: cot θ = C · d; where d is the distance between an arbitrary point to the helicoid axis, and θ is the angle between the normal and the helicoid’s axis. This rigidity property was discovered by M. Chasles in the first half of the XIXth century. Starting from this property, we give a characterization of the so-called tri-twisted metrics on the real three dimensional space with the property that a given helicoid satisfies the classical invariance condition. Similar studies can be pursued in other geometric contexts. Our most general result presents a property of surfaces of rotation observing an invariance property suggested by the analogy with Chasles’s property.

k-Sums in Abelian Groups

Given a finite subset A of an abelian group G, we study the set k ∧ A of all sums of k distinct elements of A. In this paper, we prove that |k ∧ A| ≥ |A| for all k ∈ {2,. . .,|A| − 2}, unless k ∈ {2, |A| − 2} and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite sets A ⊆ G for which |k ∧ A| = |A| for some k ∈ {2,. . .,|A| − 2}. This result answers a question of Diderrich. Our proof relies on an elementary property of proper edge-colourings of the complete graph.

Another Characterisation of Planar Graphs

A new characterisation of planar graphs is presented. It concerns the structure of the cocycle space of a graph, and is motivated by consideration of the dual of an elementary property enjoyed by sets of circuits in any graph.

Elementary properties of power series fields over finite fields

In spite of the analogies between ℚp and which became evident through the work of Ax and Kochen, an adaptation of the complete recursive axiom system given by them for ℚp, to the case of does not render a complete axiom system. We show the independence of elementary properties which express the action of additive polynomials as maps on . 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.

Suture and location of the coiling axis in gastropod shells

The general allometric equations for the logarithmic helicospiral can fit many extraconical shapes, but the isometric conditions traditionally used limit study only to conical growth. We present evidence to show that in real gastropod shells, the logarithmic helicospiral equations fit the suture. Poor location of the coiling axis and / or an inappropriate pole for the logarithmic helicospiral has often led to the rejection of this model. The differences between the errors associated with measurement or previously available models are discussed. Two methods, based on suture trace measurements, are proposed to locate the coiling axis both in apical and lateral views. The first is a graphical method based on an elementary property of the logarithmic spiral. The second is a computational method based on iterative reprojections of the suture. It is shown that the protoconch and the teleoconch must be treated separately. The precision of the new methods (especially the computing method) enables deviations from logarithmic helicospiral trajectory to be identified and differentiated from irregularities of the shell and sequential growth phases. Application of these methods may be useful not only for other gastropod morphological features, but also for other taxa such as brachiopods and other mollusks.

Bipedal reflex coordination to tactile stimulation of the sural nerve during human running

1. Cutaneous reflex responses were elicited during human running (8 km/h) on a treadmill by electrical stimulation of the sural nerve at the ankle. Stimulus trains (5 pulses of 1 ms at 200 Hz) at three nonnociceptive intensities, which were 1.5, 2.0, and 2.5 times perception threshold (PT), were delivered at 16 phases of the step cycle. For 11 subjects the surface electromyographic (EMG) activity of both the ipsilateral and contralateral long head of the biceps femoris (iBF and cBF, respectively), the semitendinosus (iST and cST), the rectus femoris (iRF and cRF), and the tibialis anterior (iTA and cTA) were recorded. 2. During human running nonnociceptive sural nerve stimulation appears to be sufficient to elicit large, widespread and statistically significant reflex responses, with a latency of approximately 80 ms and a duration of approximately 30 ms. These reflex responses seem to be an elementary property of human locomotion. This is indicated by the occurrence of the responses in all subjects, the consistency of most of the reflex patterns across the subjects and, apart from a small amount of habituation, the reproducibility of the responses during the course of the experiment. 3. The responses are modulated continuously throughout the step cycle such that their magnitude does not in general covary with the background locomotor activities. This is observed most clearly in iST, iTA, and cTA for which statistically significant reflex reversals are demonstrated, and in cRF and cTA for which the responses are gated during most of the step cycle. 4. The response magnitude generally increases as a function of increasing intensity, whereas the phase-dependent reflex modulation is intensity independent. 5. A functional dissociation within the ipsilateral hamstring muscles is demonstrated: the iBF and iST show an antagonistic reflex pattern (facilitatory and suppressive, respectively) during the periods of synergistic background locomotor activity in the step cycle. Contralaterally, however, the cBF and cST are reflexively activated as close synergists during these periods. 6. The reflex responses and their phase-dependent modulation are different for the homologous muscles in the two legs. Yet, some similarities are observed. These are present rather with respect to the phase of the corresponding leg than with respect to the phase of the stimulated leg. Both observations suggest that the phase-dependent reflex modulation is controlled separately in the ipsilateral and contralateral legs. 7. The response simultaneity in all investigated muscles supports the notion of a coordinated cutaneous interlimb reflex during human running.(ABSTRACT TRUNCATED AT 400 WORDS)

Définissabilité dans les corps de fonctions p-adiques

We study function fields over p-adically closed fields in the first-order language of fields. Using ideas of Duret [D], we show that the field of constants is definable, and that the genus is an elementary property.