Existence of meromorphic solutions of first order difference equations

It is shown that if It is shown that if \begin{equation}\label{abstract_eq} f(z+1)^n=R(z,f),\tag{\dag} \end{equation} where $R(z,f)$ is rational in $f$ with meromorphic coefficients and $\deg_f(R(z,f))=n$, has an admissible meromorphic solution, then either $f$ satisfies a difference linear or Riccati equation with meromorphic coefficients, or \eqref{abstract_eq} can be transformed into one in a list of ten equations with certain meromorphic or algebroid coefficients. In particular, if \eqref{abstract_eq}, where the assumption $\deg_f(R(z,f))=n$ has been discarded, has rational coefficients and a transcendental meromorphic solution $f$ of hyper-order $<1$, then either $f$ satisfies a difference linear or Riccati equation with rational coefficients, or \eqref{abstract_eq} can be transformed into one in a list of five equations which consists of four difference Fermat equations and one equation which is a special case of the symmetric QRT map. Solutions to all of these equations are presented in terms of Weierstrass or Jacobi elliptic functions, or in terms of meromorphic functions which are solutions to a difference Riccati equation. This provides a natural difference analogue of Steinmetz' generalization of Malmquist's theorem.Comment: 25 pages, some errors are correcte

On meromorphic solutions of a second order differential equation proposed by Hayman

This paper mainly consists of two parts. First, we use the idea of Halburd and Wang to give the form of all admissible meromorphic solutions of the second order differential equation: $w''w-w'^2+a ww'+b w^2=\alpha w+\beta w'+\gamma$, where $a,b,\alpha,\beta,\gamma$ are small meromorphic functions of $w$ in the sense of Nevanlinna. This allows us to find all transcendental meromorphic solutions of an equation proposed by Hayman: $a_1(f''f-f'^2)+a_2f'f+a_3f^2+b_1f''+b_2f'+b_3f+b_4=0$, $(\dag)$ where $a_i$ and $b_j$ are polynomials, all of which have finite hyper order at most. Second, we consider an autonomous equation related to $(\dag)$, namely: $f''f=p_1f'^2+p_2f'f+p_3f'+p_4f^4+p_5f^3+p_6f^2+p_7f+p_8$, $(\ddag)$ where $p_1\in\{\frac{1}{2},1,2,\frac{3}{2},\cdots,\frac{m+1}{m},\cdots\}$, $m\in \mathbb{Z}^{+}$ and $p_2,\cdots,p_8$ are constants. We use the Ahlfors--Shimizu characteristic to prove that all meromorphic solutions of $(\ddag)$ have hyper order not exceeding~$2$. The method also yields that all meromorphic solutions of the third Painlev\'{e} equation $P_{III}$ have order not exceeding~$4$.Comment: 17 page

Effective preparation and collisional decay of atomic condensate in excited bands of an optical lattice

We present a method for the effective preparation of a Bose-Einstein condensate (BEC) into the excited bands of an optical lattice via a standing-wave pulse sequence. With our method, the BEC can be prepared in either a single Bloch state in a excited-band, or a coherent superposition of states in different bands. Our scheme is experimentally demonstrated by preparing a $^{87}$Rb BEC into the $d$-band and the superposition of $s$- and $d$-band states of a one-dimensional optical lattice, within a few tens of microseconds. We further measure the decay of the BEC in the $d$-band state, and carry an analytical calculation for the collisional decay of atoms in the excited-band states. Our theoretical and experimental results consist well.Comment: 9 pages, 5 figures, Accepted by Phys. Rev.

Neuroimaging investigations of cortical specialisation for different types of semantic knowledge

Embodied theories proposed that semantic knowledge is grounded in motor and perceptual experiences. This leads to two questions: (1) whether the neural underpinnings of perception are also necessary for semantic cognition; (2) how do biases towards different sensorimotor experiences cause brain regions to specialise for particular types of semantic information. This thesis tackles these questions in a series of neuroimaging and behavioural investigations. Regarding question 1, strong embodiment theory holds that semantic representation is reenactment of corresponding experiences, and brain regions for perception are necessary for comprehending modality-specific concepts. However, the weak embodiment view argues that reenactment may not be necessary, and areas near to perceiving regions may be sufficient to support semantic representation. In the particular case of motion concepts, lateral occipital temporal cortex (LOTC) has been long identified as an important area, but the roles of its different subregions are still uncertain. Chapter 3 examined how different parts of LOTC reacted to written descriptions of motion and static events, using multiple analysis methods. A series of anterior to posterior sub-regions were analyzed through univariate, multivariate pattern analysis (MVPA), and psychophysical interaction (PPI) analyses. MVPA revealed strongest decoding effects for motion vs. static events in the posterior parts of LOTC, including both visual motion area (V5) and posterior middle temporal gyrus (pMTG). In contrast, only the middle portion of LOTC showed increased activation for motion sentences in univariate analyses. PPI analyses showed increased functional connectivity between posterior LOTC and the multiple demand network for motion events. These findings suggest that posterior LOTC, which overlapped with the motion perception V5 region, is selectively involved in comprehending motion events, while the anterior part of LOTC contributes to general semantic processing. Regarding question 2, the hub-and-spoke theory suggests that anterior temporal lobe (ATL) acts as a hub, using inputs from modality-specific regions to construct multimodal concepts. However, some researchers propose temporal parietal cortex (TPC) as an additional hub, specialised in processing and integrating interaction and contextual information (e.g., for actions and locations). These hypotheses are summarized as the "dual-hub theory" and different aspects of this theory were investigated in in Chapters 4 and 5. Chapter 4 focuses on taxonomic and thematic relations. Taxonomic relations (or categorical relations) occur when two concepts belong to the same category (e.g., ‘dog’ and ‘wolf’ are both canines). In contrast, thematic relations (or associative relations) refer to situations that two concepts co-occur in events or scenes (e.g., ‘dog’ and ‘bone’), focusing on the interaction or association between concepts. Some studies have indicated ATL specialization for taxonomic relations and TPC specialization for thematic relations, but others have reported inconsistent or even converse results. Thus Chapter 4 first conducted an activation likelihood estimation (ALE) meta-analysis of neuroimaging studies contrasting taxonomic and thematic relations. This found that thematic relations reliably engage action and location processing regions (left pMTG and SMG), while taxonomic relations only showed consistent effects in the right occipital lobe. A primed semantic judgement task was then used to test the dual-hub theory’s prediction that taxonomic relations are heavily reliant on colour and shape knowledge, while thematic relations rely on action and location knowledge. This behavioural experiment revealed that action or location priming facilitated thematic relation processing, but colour and shape did not lead to priming effects for taxonomic relations. This indicates that thematic relations rely more on action and location knowledge, which may explain why the preferentially engage TPC, whereas taxonomic relations are not specifically linked to shape and colour features. This may explain why they did not preferentially engage left ATL. Chapter 5 concentrates on event and object concepts. Previous studies suggest ATL specialization for coding similarity of objects’ semantics, and angular gyrus (AG) specialization for sentence and event structure representation. In addition, in neuroimaging studies, event semantics are usually investigated using complex temporally extended stimuli, unlike than the single-concept stimuli used to investigate object semantics. Thus chapter 5 used representational similarity analysis (RSA), univariate analysis, and PPI analysis to explore neural activation patterns for event and object concepts presented as static images. Bilateral AGs encoded semantic similarity for event concepts, with the left AG also coding object similarity. Bilateral ATLs encoded semantic similarity for object concepts but also for events. Left ATL exhibited stronger coding for events than objects. PPI analysis revealed stronger connections between left ATL and right pMTG, and between right AG and bilateral inferior temporal gyrus (ITG) and middle occipital gyrus, for event concepts compared to object concepts. Consistent with the meta-analysis in chapter 4, the results in chapter 5 support the idea of partial specialization in AG for event semantics but do not support ATL specialization for object semantics. In fact, both the meta-analysis and chapter 5 findings suggest greater ATL involvement in coding objects' associations compared to their similarity. To conclude, the thesis provides support for the idea that perceptual brain regions are engaged in conceptual processing, in the case of motion concepts. It also provides evidence for a specialised role for TPC regions in processing thematic relations (pMTG) and event concepts (AG). There was mixed evidence for specialisation within the ATLs and this remains an important target for future research

The Influence of Face Inversion and Spatial Frequency on the Self-Positive Expression Processing Advantage

Previous research has examined the impact of late self-evaluation, ignoring the impact of the early visual coding stage and the extraction of facial identity information and expression information on the self-positive expression processing advantage. From the perspective of the processing course, this study examined the stability of the self-positive expression processing advantage and revealed its generation mechanism. In Experiment 1, inverted self-expression and others’ expressive pictures were used to influence early structural coding. In Experiments 2a and 2b, we used expression pictures of high and low spatial frequency, thereby affecting the extraction of facial identity information or expression information in the mid-term stage. The visual search paradigm was adopted in three experiments, asking subjects to respond to the target expression. We found that under the above experimental conditions, the search speed for self-faces was always faster than that for self-angry expressions and others’ faces. These results showed that, compared with others’ expressions and self-angry expressions, self-positive expressions were more prominent and more attractive. These findings suggest that self-expression recognition combines with conceptual self-knowledge to form an abstract and constant processing pattern. Therefore, the processing of self-expression recognition was not affected by the facial orientation and spatial frequencies

Representation of event and object concepts in ventral anterior temporal lobe and angular gyrus

Semantic knowledge includes understanding of objects and their features and also understanding of the characteristics of events. The hub-and-spoke theory holds that these conceptual representations rely on multiple information sources that are integrated in a central hub in the ventral anterior temporal lobes. The dual-hub theory expands this framework with the claim that the ventral anterior temporal lobe hub is specialized for object representation, while a second hub in angular gyrus is specialized for event representation. To test these ideas, we used representational similarity analysis, univariate and psychophysiological interaction analyses of fMRI data collected while participants processed object and event concepts (e.g. “an apple,” “a wedding”) presented as images and written words. Representational similarity analysis showed that angular gyrus encoded event concept similarity more than object similarity, although the left angular gyrus also encoded object similarity. Bilateral ventral anterior temporal lobes encoded both object and event concept structure, and left ventral anterior temporal lobe exhibited stronger coding for events. Psychophysiological interaction analysis revealed greater connectivity between left ventral anterior temporal lobe and right pMTG, and between right angular gyrus and bilateral ITG and middle occipital gyrus, for event concepts compared to object concepts. These findings support the specialization of angular gyrus for event semantics, though with some involvement in object coding, but do not support ventral anterior temporal lobe specialization for object concepts