We consider the dilation property of the modulation spaces M p,q . Let D λ : f (t) → f (λt) be the dilation operator, and we consider the behavior of the operator norm D λ M p,q →M p,q with respect to λ. Our result determines the best order for it, and as an application, we establish the optimality of the inclusion relation between the modulation spaces and Besov spaces, which was proved by Toft [J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus, I, J. Funct. Anal. 207 (2004) 399-429].
Inflation is the leading theory of the first instant of the universe. Inflation, which postulates that the universe underwent a period of rapid expansion an instant after its birth, provides convincing explanation for cosmological observations. Recent advancements in detector technology have opened opportunities to explore primordial gravitational waves generated by the inflation through "B-mode" (divergent-free) polarization pattern embedded in the Cosmic Microwave Background anisotropies. If detected, these signals would provide strong evidence for inflation, point to the correct model for inflation, and open a window to physics at ultra-high energies.LiteBIRD is a satellite mission with a goal of detecting degree-and-larger-angular-scale B-mode polarization. Lite-BIRD will observe at the second Lagrange point with a 400 mm diameter telescope and 2,622 detectors. It will survey the entire sky with 15 frequency bands from 40 to 400 GHz to measure and subtract foregrounds.The U.S. LiteBIRD team is proposing to deliver sub-Kelvin instruments that include detectors and readout electronics. A lenslet-coupled sinuous antenna array will cover low-frequency bands (40 GHz to 235 GHz) with four frequency arrangements of trichroic pixels. An orthomode-transducer-coupled corrugated horn array will cover high-frequency bands (280 GHz to 402 GHz) with three types of single frequency detectors. The detectors will be made with Transition Edge Sensor (TES) bolometers cooled to a 100 milli-Kelvin base temperature by an adiabatic demagnetization refrigerator. The TES bolometers will be read out using digital frequency multiplexing with Superconducting QUantum Interference Device (SQUID) amplifiers. Up to 78 bolometers will be multiplexed with a single SQUID amplier.We report on the sub-Kelvin instrument design and ongoing developments for the LiteBIRD mission.
In this review, we have tried to elucidate substrates for the execution of normal gait and to understand pathophysiological mechanisms of gait failure in basal ganglia dysfunctions. In Parkinson's disease, volitional and emotional expressions of movement processes are seriously affected in addition to the disturbance of automatic movement processes, such as adjustment of postural muscle tone before gait initiation and rhythmic limb movements during walking. These patients also suffer from muscle tone rigidity and postural instability, which may also cause reduced walking capabilities in adapting to various environments. Neurophysiological and clinical studies have suggested the importance of basal ganglia connections with the cerebral cortex and limbic system in the expression of volitional and emotional behaviors. Here we hypothesize a crucial role played by the basal ganglia-brainstem system in the integrative control of muscle tone and locomotion. The hypothetical model may provide a rational explanation for the role of the basal ganglia in the control of volitional and automatic aspects of movements. Moreover, it might also be beneficial for understanding pathophysiological mechanisms of basal ganglia movement disorders. A part of this hypothesis has been supported by studies utilizing a constructive simulation engineering technique that clearly shows that an appropriate level of postural muscle tone and proper acquisition and utilization of sensory information are essential to maintain adaptable bodily functions for the full execution of bipedal gait. In conclusion, we suggest that the major substrates for supporting bipedal posture and executing bipedal gait are 1) fine neural networks such as the cortico-basal ganglia loop and basal ganglia-brainstem system, 2) fine musculoskeletal structures with adequately developed (postural) muscle tone, and 3) proper sensory processing. It follows that any dysfunction of the above sensorimotor integration processes would result in gait disturbance.
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