In history, theta function identities have been studied extensively and deeply. In this paper, applying the residue theorem of an elliptic function, we obtain a parameterized theta function identity. Then, by selecting the appropriate parameter, we derive some famous Jacobi identities, as well as other relatively simple theta function identities.

This article provides a proof that the Ramanujan's Inequality given by, \begin{align*} \pi(x)^2 < \frac{e x}{\log x} \pi\left(\frac{x}{e}\right) \end{align*} holds unconditionally for every $x\geq \exp(43.5102146)$. In case for an alternate proof of the result stated above, we shall exploit certain estimates involving the Chebyshev Theta Function, $\vartheta(x)$ in order to derive appropriate bounds for $\pi(x)$, which'll lead us to a much improved condition for the inequality proposed by Ramanujan to satisfy unconditionally.

Jacobian theta function identities are related to the Addition formula for the Weierstrass 1 Sigma function. In this paper, from the Addition formula for Weierstrass Sigma function, new and 2 simple proofs of four square sums theorem are given. Using the addition formula of the theta func- 3 tions, we also give simple proofs of two square sums and eight square sums theorem.

We establish certain modular equations of degree three based on some results of Ramanujan's theta functions.

The dentate gyrus (DG) is part of the hippocampal formation and is essential for important cognitive processes such as navigation and memory. The oscillatory activity of the DG network is believed to play a critical role in cognition. DG circuits generate three main rhythms: theta, beta, and gamma, which participate in the specific information processing performed by DG neurons. In the temporal lobe epilepsy (TLE), cognitive abilities are impaired, which may be due to drastic alterations in the DG structure and network activity during epileptogenesis. The theta rhythm and theta coherence are especially vulnerable in dentate circuits; disturbances in DG theta oscillations and their coherence may be responsible for general cognitive impairments observed during epileptogenesis. Some researchers suggested that the vulnerability of DG mossy cells is a key factor in the genesis of TLE, but others did not support this hypothesis. The aim of the review is not only to present the current state of the art in this field of research but to help pave the way for future investigations by highlighting the gaps in our knowledge to completely appreciate the role of DG rhythms in brain functions. Disturbances in oscillatory activity of the DG during TLE development are described in detail that may be a diagnostic marker in the treatment of this disease.

The brain rhythms are essential for information processing in neuronal networks. Oscillations recorded in different brain regions can be synchronized and have a constant phase difference, i.e. be coherent. Coherence between local field potential (LFP) signals from different regions in the brain may be correlated with the performance of cognitive tasks, from which it is concluded that these regions of the brain are involved in the task performance together. In this review, we discuss why coherence occurs and how it is coupled to the information transfer between different regions of the hippocampal formation. Coherence in theta and gamma frequency ranges is described since these rhythms are most pronounced during the hippocampus-dependent attention and memory. We review in vivo studies of interactions between different regions of the hippocampal formation in theta and gamma frequency bands. The kay provisions of the review: 1) coherence emerges from synchronous postsynaptic currents in principal neurons, occurring as a result of synchronization of neuronal spike activity; 2) synchronization of neuronal spike patterns in two regions of the hippocampal formation can be realised through induction or resonance; 3) coherence at a specific time point reflects the transfer of information between regions of the hippocampal formation, in particular, gamma coherence reflects the coupling of active neuronal ensembles. Overall, coherence is not an epiphenomenon, but an important physiological process that has certain generation mechanisms and performs important functions in information processing and transmission across the brain regions.

DNA polymerases replicate cell genomes and/or participate in the maintenance of genome integrity. DNA polymerases sharing high sequence homology with E. coli DNA polymerase I (pol I), have been grouped in Family A. Pol I participates in Okazaki fragment maturation and in bacterial genome repair. Since its discovery in 1956, pol I has been extensively studied, primarily to get deeper insights into the mechanism of DNA replication. As research on DNA polymerases advances, many novel functions of this group of polymerases are uncovered. For example, human DNA polymerase θ (a Family A DNA pol) has been shown to synthesize DNA using RNA as a template, a function typically attributed to retroviral reverse transcriptase. A heightened interest in drug discovery against pol θ has emerged due to its roles in cancer. Likewise, Pol I family enzymes also appear attractive drug-development targets against microbial infections. Development of antimalarial compounds targeting apicoplast apPOL, an ortholog of Pol I, further extends the druggability of this family of enzymes. Here, we summarize reported drug-development efforts against Family A polymerases, and future perspective regarding these enzymes as antibiotic targets. Current techniques such artificial intelligence can be used to facilitate the goal of new drugs.

Hippocampus plays a crucial role in spatial and episodic memory. The acquisition of new memories is impossible without participation of the hippocampus. There are two main functional states or “modes” of the hippocampal activity, theta and non-theta state. They have different behavioral correlates and clearly different spectral content of the LFPs and neuronal spiking. Hippocampal theta state is present under active exploratory behavior, locomotion, cognitive situations requiring attention and REM sleep. Slow-wave sleep and quiet wakefulness (immobility, eating, grooming) represent the non-theta hippocampal state. The nodal point for the regulation of hippocampal activity is the MS. The activation or suppression of different types of MS cells appears to be responsible for controlling hippocampal theta and non-theta states. Functional coupling between MS neurons and hippocampal interneurons varies in a state-dependent manner.

Yogic and meditative traditions have long held that the fluctuations of the breath and the mind are intimately related. While respiratory modulation of cortical activity and attentional switching are established, the extent to which electrophysiological markers of attention exhibit synchronization with respiration is unknown. To this end, we examined 1) frontal midline theta-beta ratio, an indicator of attentional control state known to correlate with mind wandering episodes and functional connectivity of the executive control network; 2) pupil diameter (PD), a known proxy measure of locus coeruleus (LC) noradrenergic activity; and 3) respiration for evidence of phase synchronization and information transfer (multivariate Granger causality) during quiet restful breathing. Our results indicate that both TBR and PD are simultaneously synchronized with the breath, suggesting an underlying oscillation of an attentionally relevant electrophysiological index that is phase-locked to the respiratory cycle which could have the potential to bias the attentional system into switching states. We highlight the LC’s pivotal role as a coupling mechanism between respiration and TBR, and elaborate on its dual functions as both a chemosensitive respiratory nucleus and a pacemaker of the attentional system. We further suggest that an appreciation of the dynamics of this weakly coupled oscillatory system could help deepen our understanding of the traditional claim of a relationship between breathing and attention.

Cortical theta burst stimulation (TBS) structured as intermittent (iTBS) and continuous (cTBS) could prevent the progression of the experimental autoimmune encephalomyelitis (EAE). The interplay of brain antioxidant defense systems against overproduction of reactive oxygen, nitrogen, and thiol species induced by EAE has not been entirely investigated, just as the effect of iTBS or cTBS on oxidative-nitrogen stress (ONS) in EAE rats. Dark Agouti strain female rats were tested for the effects of EAE and TBS. The rats were randomly divided into the following groups: C - control, EAE - rats immunized for EAE, CFA - rats immunized with Complete Freund's adjuvant; iTBS and cTBS groups, and EAE+iTBS and EAE+cTBS - health and EAE rats exposed to iTBS and cTBS, respectively; EAE+iTBSsh and EAE+cTBSsh - sham stimulated EAE rats with the same noise artifacts of iTBS and cTBS, respectively. Superoxide dismutase activity, levels of superoxide anion (O2•-), lipid peroxidation, glutathione (GSH), nicotinamide adenine dinucleotide phosphate (NADPH) and thioredoxin reductase (TrxR) activity were analyzed in rat spinal cords homogenates. The severity of EAE clinical coincided with the climax of ONS, based on the increase of superoxide anion and lipid peroxidation; depletion of total thiols, GSH and NADPH; and decrease of SOD activity. The TrxR imposed the most sensitive response against the applied central nervous system (CNS) stressors to rats. We concluded that the TrxR upregulation meritoriously compensates decreased ROS sequestrating and GSH systems in EAE. Both iTBS and cTBS modulate the biochemical environment at a distance from the area of stimulation against ONS, accomplish a similar effect on TrxR activity to EAE and healthy rats, and alleviate symptoms of EAE.

Theta oscillations generated in hippocampal (HPC) and cortical neuronal networks are involved in various aspects of brain function, including sensorimotor integration, movement planning, memory formation and attention. Disruptions of theta rhythms are present in individuals with various disorders, including epilepsy and Alzheimer’s disease. Theta rhythm generation involves a specific interplay between cellular (ionic) and network (synaptic) mechanisms. HCN channels are theta modulators, and several medications are known to enhance their activity. We investigated how different doses of lamotrigine (LTG), an HCN channel activator, and antiepileptic and neuroprotective agent, would affect hippocampal theta rhythms in acute HPC slices (in vitro) and anaesthetized rats (in vivo). Whole-cell patch clamp recordings revealed that LTG decreased GABAA-fast transmission in CA3 and CA1 cells, in vitro. In addition, LTG directly depressed CA3 and CA1 pyramidal neuron excitability. These effects were partially blocked by ZD 7288, a selective HCN blocker, and are consistent with decreased excitability associated with antiepileptic actions. Lamotrigine also depressed hippocampal theta oscillations in vitro, also consistent with its neuronal depressant effects. In contrast, it exerted an opposite, enhancing effect, on theta recorded in vivo. The contradictory in vivo and in vitro results indicate that LTG increases ascending theta activating medial septum/entorhinal synaptic inputs that over-power the depressant effects seen in hippocampal neurons. These results provide new insights into LTG actions and indicate an opportunity to develop more precise therapeutics for the treatment of dementias, memory disorders and epilepsy.

Objective: This study aimed to explore longitudinal relationships between neurophysiological biomarkers and upper limb motor function recovery in stroke patients, focusing on electroencephalography (EEG) and transcranial magnetic stimulation (TMS) metrics. Methods: This longitudinal cohort study analyzed neurophysiological, clinical, and demographic data from 102 stroke patients enrolled in the DEFINE cohort. We investigated the associations between baseline and post-intervention changes in EEG Theta/Alpha Ratio (TAR) and TMS metrics with upper limb motor functionality, assessed by five outcomes: Fugl-Meyer Assessment (FMA), Handgrip Strength Test (HST), Pinch Strength Test (PST), Finger Tapping Test (FTT), and Nine-Hole Peg Test (9HPT). Results: Our multivariate models identified that a higher baseline TAR in the lesioned hemisphere was consistently associated with poorer motor outcomes across all five assessments. Conversely, a higher improvement in TAR was positively associated with improvements in FMA and 9HPT. Additionally, increased TMS Motor Evoked Potential (MEP) amplitude in the non-lesioned hemisphere correlated with greater FMA-diff, while lower TMS Intracortical Inhibition (ICI) in the non-lesioned hemisphere was linked to better PST improvements. These findings suggest the potential of TAR and TMS metrics as biomarkers for predicting motor recovery in stroke patients. Conclusion: Our findings highlight the significance of TAR in the lesioned hemisphere as a predictor of motor function recovery post-stroke and also a potential signature for compensatory oscillations. The observed relationships between TAR and motor improvements, as well as the associations with TMS metrics, underscore the potential of these neurophysiological measures in guiding personalized rehabilitation strategies for stroke patients.

This investigation introduces an efficient method for assessing the reliability of semi-rigid steel frame connections in existing buildings, explicitly targeting the Top Angle and Seat Pad (TA-SP) semi-rigid connection commonly used in Australian cities' high-rise buildings. The moment capacity of these connections was examined through parameter power model testing, meticulously considering the bending, shear, and prying forces actions to determine the weakest elements in the connection system. A plastic hinge model was applied to ensure computational efficiency and accuracy in assessing the connections' ultimate strength. Comparatively, the finite element (F.E) method M−θ curves were introduced to scrutinise the nonlinear behaviour of the connection, providing validation for the predicted analytical approach. Further emphasis was placed on identifying the reliability of the weakest elements in the connection system, employing the First-Order and Second-Order Reliability Methods models to estimate their reliability index. The correlation between the model's failure probability and a percentile error was also explored, utilising Monte Carlo simulation reference for refined validation. Novel formulas and methodologies were introduced to compute the locked-chain-series system reliability, which accounts for the inherent material and geometric variability. The findings highlight the critical significance of the weakest element in determining the overall reliability of the existing TA-SP connection.

Terahertz tomography is a promising method among non-destructive inspection techniques to detect faults and defects in dielectric samples. Image quality recently significantly improved through the incorporation of a priori information and off-axis data. However, this improvement has come at the cost of increased measurement time. To aim toward industrial applications, it is therefore necessary to speed up the measurement by parallelizing the data acquisition employing multi-channel setups. In this work, we present two tomographic frequency-modulated continuous wave (FMCW) systems working at a bandwidth of 230-320 GHz, equipped with an eight-channel detector array, and compare their imaging results with a single-pixel setup. While in the first system, the additional channels are used exclusively to detect radiation refracted by the sample, the second system features an f-θ lens, focusing the beam at different positions on its flat focal plane, and thus utilizing the whole detector array directly. The usage of the f-θ lens in combination with a scanning mirror eliminates the necessity of the formerly used slow translation of a single-pixel transmitter. This opens up the potential for a significant increase in acquisition speed, in our case by a factor of four to five, respectively.