This thesis attempts to establish properties of Hölder and Cesàro summable series analogous to those of ordinary convergent series and also to establish properties that are possibly different from those of convergent series.

Some of the properties of the specific summation methods will be investigated, such as what type of divergent series a method can or cannot sum, if the insertion of zeros into a series does change the sum, and when different methods give the same sum for a series.

Secure multiparty computation refers to the problem where a number of users wish to securely compute a function on their inputs without revealing any unnecessary information. This dissertation focuses on the fundamental limits of secure summation under different constraints. We first focus on the minimal model of secure computation, in which two users each hold an input and wish to securely compute a function of their inputs at the server. We propose a novel scheme base on the algebraic structure of finite field and modulo ring of integers. Then we extend the minimal model of secure computation, in which K users wish to securely compute the sum of their inputs at the server. We prove a folklore result on the limits of communication cost and randomness cost. Then we characterized the optimal communication cost with user dropouts constraint, when some users may lose connection to the server and the server wishes to compute the sum of remaining inputs. Next, we characterize the optimal communication and randomness cost for symmetric groupwise keys and find the feasibility condition for arbitrary groupwise keys. Last, we study the secure summation with user selection, such that the server may select any subset of users to compute the sum of their inputs. This leads us to the MDS variable generation problem. We characterize the optimal individual key rate and the result is interestingly the harmonic number.

There was few human study evaluated the analgesic effect of acupuncture on central nervous system (CNS). The electrical temporal summation (TS) pain model has been validated and provides the opportunity to study the central inhibition effect of acupuncture in healthy humans. The present study aimed to: 1. systematically review available randomised, controlled trials (RCTs) of acupuncture on experimentally induced pain in healthy humans; 2. conduct a RCT to assess the effect of manual acupuncture (MA) and electro-acupuncture (EA) on TS of pain and the spatial (i.e. the local and remote sites to acupuncture stimulation) and the temporal (i.e. immediately after and 24-hours after the intervention) characteristics of this effect. The systematic review was carried out in accordance with the requirements of a Cochrane Systematic Review. The methodological quality and credibility of the acupuncture intervention of the included RCTs were assessed. The Review Management software (RevMan version 4.2, The Cochrane Library) was used for data extraction and data analysis. 605 papers were identified from four databases (Pubmed, Cochrane Library, CINAHL and EMBASE). Only nine papers met the inclusion criteria. The methodological quality and credibility of the acupuncture invention were satisfactory. The pain models and interventions applied varied substantially from study to study. Consequently, meta-analyses were not conducted. Comparing acupuncture with non-invasive control, significant acupuncture analgesia was reported. These studies also demonstrated that invasive controls produced analgesia. For the RCT of acupuncture on TS, 27 healthy volunteers were recruited and randomly assigned to either EA, MA or sham-acupuncture (SA) group, with nine volunteers in each group. To test pain thresholds, transcutaneous electrical stimulation was delivered to two sites on the anterior aspects of both legs and one site on the dorsum of the non-dominant forearm. Pain thresholds to single electrical stimulation (SPT) and to TS stimulation (TST) were assessed before, 30-minutes after and 24-hours after the intervention. Acupuncture was given to Zusanli (ST36) and Fenglong (ST 40) on the dominant leg. The level of anxiety was assessed before and after acupuncture with Spielberg State and Anxiety Inventory. The three groups were comparable at baseline. The level of anxiety did not change significantly after acupuncture. EA significantly increased SPT and TST on the treatment leg 24-hour after the treatment when compared with SA (p less than 0.05), but did not increase those measured on the non-treatment leg or the forearm. The fact that such an effect increased within 24 hours after acupuncture might indicate the potential role of neurohumoral mechanisms in acupuncture analgesia. The analgesia effect of EA on TS tended to be localised at the needling site. This observation is different from the understanding of the wide-spread effect of acupuncture. The discrepancy could be due to the small sample size of the current study. In conclusion, this is the first study that demonstrates EA elicits a strong inhibition on the CNS in health humans. Such a central effect lasts more than 24 hours, and limits to the site where acupuncture is applied. These findings need to be confirmed in other TS models.

Although lymphatic vessel behavior is analogous to hearts (e.g. systole and diastole) and blood vessels (e.g. basal tone), hearts and blood vessels have fundamentally different contractile properties. While summation during contraction is minimized in the heart, summation is necessary for tonic contraction in blood vessels. Because lymphatic vessel behavior mimics cardiac and vascular behavior, we hypothesized that above a critical contraction frequency there is significant summation, evidenced by significantly increased diastolic active tension (i.e. basal tone). We used an isovolumic, controlled-flow preparation to examine the interaction of contraction cycle-time with contraction frequency. Using segments of isolated lymphatic vessels (~1 cm in length and 3-4 mm in diameter) from bovine mesentery, we measured transmural pressure and diameter for end-diastole and end-systole during spontaneous contractions for 10 volume steps. We found time between contractions (beat-to-beat period) decreases with increasing diameter, and total contraction time (vessel twitch length, 11.08 ± 1.54 s) slightly increases with increasing diameter. At the intersection of these relationships, there is a critical period, below which the vessel does not have time to fully relax. Above the diameter at the critical period, diastolic active tension (end-diastolic minus passive vessel tension) significantly increases with increases in diameter (309 to 562% change in slope, p<0.0001), and, below the critical period, diastolic active tension increases with decreases in beat-to-beat period (712 to 2208% change in slope, p<0.0014). Because this transition occurs within a physiological range, it suggests summation may be crucial for lymphatic vessel function as a pump and a conduit.

Although lymphatic vessel behavior is analogous to hearts (e.g. systole and diastole) and blood vessels (e.g. basal tone), hearts and blood vessels have fundamentally different contractile properties. While summation during contraction is minimized in the heart, summation is necessary for tonic contraction in blood vessels. Because lymphatic vessel behavior mimics cardiac and vascular behavior, we hypothesized that above a critical contraction frequency there is significant summation, evidenced by significantly increased diastolic active tension (i.e. basal tone). We used an isovolumic, controlled-flow preparation to examine the interaction of contraction cycle-time with contraction frequency. Using segments of isolated lymphatic vessels (~1 cm in length and 3-4 mm in diameter) from bovine mesentery, we measured transmural pressure and diameter for end-diastole and end-systole during spontaneous contractions for 10 volume steps. We found time between contractions (beat-to-beat period) decreases with increasing diameter, and total contraction time (vessel twitch length, 11.08 ± 1.54 s) slightly increases with increasing diameter. At the intersection of these relationships, there is a critical period, below which the vessel does not have time to fully relax. Above the diameter at the critical period, diastolic active tension (end-diastolic minus passive vessel tension) significantly increases with increases in diameter (309 to 562% change in slope, p<0.0001), and, below the critical period, diastolic active tension increases with decreases in beat-to-beat period (712 to 2208% change in slope, p<0.0014). Because this transition occurs within a physiological range, it suggests summation may be crucial for lymphatic vessel function as a pump and a conduit.

Thesis (M.A.)--University of Hawaii at Manoa, 2004. / Includes bibliographical references (leaves 47-48). / v, 48 leaves, bound ill. 29 cm

<p> In six conditioned suppression experiments with rats, two conditioned stimuli (CSs) were individually trained and then tested as a compound. In one set of experiments, the suppressing effect of the compound was greater than that of either CS presented alone. This result is referred to as compound summation. In a second set of experiments, the suppressing effect of the compound was less than that of the "stronger" suppressing individual CS. This result is referred to as compound attenuation. The combination of summation and attenuation makes it possible to determine whether CSs with unknown properties are weakly excitatory (i.e., weak suppressors) or inhibitory (i.e., conditioned characteristics that are opposite the excitatory suppressing effect). If an unknown CS is tested in compound with a second CS known to be excitatory, summation indicates that the unknown stimulus is excitatory, while attenuation indicates that the unknown stimulus is inhibitory. In a final set of experiments, this compound test procedure was used to examine extinction and differential conditioning as inhibitory training procedures. Extensive extinction of a previously trained CS, even far beyond the point at which suppression vanished, was found to be an ineffective inhibitory training procedure. Rather, compound tests showed that the stimulus retained excitatory properties. Differential conditioning was found to be a very effective inhibitory training procedure, regardless of whether presentations of a previously trained CS and shock, shook alone, or the previously trained CS - alone accompanied the unreinforced CS undergoing inhibitory conditioning. These findings are discussed in terms of current theories of conditioning and unresolved issues surrounding the acquisition and maintenance of inhibitory properties.</p> / Thesis / Doctor of Philosophy (PhD)

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Pain is a remarkably complex and a multifaceted process, involving the interaction between physiological and psychological factors in unique ways. Among many other factors, the size of the affected surface area contributes to the pain experience, altering one’s pain perception. Spatial summation is the term used to describe this phenomenon, and is characterized by an increase in pain perception, or a decrease in pain threshold, when the affected surface area is increased. This project investigated the neuronal processes underlying spatial summation of heat sensations in healthy female volunteers, by means of functional magnetic resonance imaging (fMRI) of the central nervous system. The first study of this project involved increasing the surface area of skin stimulated by manipulating the number of thermal probes delivering thermal stimulation, which was delivered just below participants’ measured pain threshold. Surface area was increased on one hand and across two hands to determine the extent of spatial summation, and furthermore, to determine the effect on neural activity in the spinal cord and brainstem. The second study of this project involved increasing the surface area of skin stimulated by a noxious heat stimulus and its effect on pain perception and corresponding neural activity in the spinal cord, brainstem, and brain. Results from this project suggest that the central mechanisms contributing to the spatial summation of heat sensations involve many of many of the brainstem and brain regions involved in processing the emotional, motivational, and cognitive aspects of pain. Therefore, increasing the surface area of stimulation may alter pain perception by influencing the affective dimension of the sensation, rather than the sensory/discriminatory component. The combination of such structures may interact in a unique way to protect the body from potential, or further damage, by increasing the perception of pain through emotional, motivational and cognitive mechanisms. / Thesis (Master, Neuroscience Studies) -- Queen's University, 2013-07-15 19:01:37.694