Probability Laws Derived from the Gamma Function

Several densities or probability laws of continuous random variables derive from the Euler Gamma function. These laws form the basis of sampling theory, namely hypothesis testing and estimation. Namely the gamma, beta, and Student law, through the chi-square law and the normal law are all distributions resulting from applications of Euleur functions.

Evaluation of Generalized Error Function via Fast-Converging Power Series

A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.

SFC placement and dynamic resource allocation based on VNF performance-resource function and service requirement in cloud-edge environment

With the continuous development of network func-tions virtualization(NFV)and software-defined networking(SDN)technologies and the explosive growth of network traffic,the requirement for computing resources in the network has risen sharply.Due to the high cost of edge computing resources,coordinating the cloud and edge computing resources to improve the utilization efficiency of edge computing resources is still a considerable challenge.In this paper,we focus on optimiz-ing the placement of network services in cloud-edge environ-ments to maximize the efficiency.It is first proved that,in cloud-edge environments,placing one service function chain(SFC)integrally in the cloud or at the edge can improve the utilization efficiency of edge resources.Then a virtual network function(VNF)performance-resource(P-R)function is proposed to repre-sent the relationship between the VNF instance computing per-formance and the allocated computing resource.To select the SFCs that are most suitable to deploy at the edge,a VNF place-ment and resource allocation model is built to configure each VNF with its particular P-R function.Moreover,a heuristic recur-sive algorithm is designed called the recursive algorithm for max edge throughput(RMET)to solve the model.Through simula-tions on two scenarios,it is verified that RMET can improve the utilization efficiency of edge computing resources.

A Set of Almost Automorphic Functions and Applications

For a set S of real numbers, we introduce the concept of S-almost automorphic functions valued in a Banach space. It generalizes in particular the space of Z-almost automorphic functions. Considering the space of S-almost automorphic functions, we give sufficient conditions of the existence and uniqueness of almost automorphic solutions of a differential equation with a piecewise constant argument of generalized type. This is done using the Banach fixed point theorem.

Brain Functional Network Changes in Patients with Poststroke Cognitive Impairment Following Acupuncture Therapy

Background: The mechanisms by which acupuncture affects poststroke cognitive impairment (PSCI) remain unclear. Objective: To investigate brain functional network (BFN) changes in patients with PSCI after acupuncture therapy. Methods: Twenty-two PSCI patients who underwent acupuncture therapy in our hospital were enrolled as research subjects. Another 14 people matched for age, sex, and education level were included in the normal control (HC) group. All the subjects underwent resting-state functional magnetic resonance imaging (rs-fMRI) scans;the PSCI patients underwent one scan before acupuncture therapy and another after. The network metric difference between PSCI patients and HCs was analyzed via the independent-sample t test, whereas the paired-sample t test was employed to analyze the network metric changes in PSCI patients before vs. after treatment. Results: Small-world network attributes were observed in both groups for sparsities between 0.1 and 0.28. Compared with the HC group, the PSCI group presented significantly lower values for the global topological properties (γ, Cp, and Eloc) of the brain;significantly greater values for the nodal attributes of betweenness centrality in the CUN. L and the HES. R, degree centrality in the SFGdor. L, PCG. L, IPL. L, and HES. R, and nodal local efficiency in the ORBsup. R, ORBsupmed. R, DCG. L, SMG. R, and TPOsup. L;and decreased degree centrality in the MFG. R, IFGoperc. R, and SOG. R. After treatment, PSCI patients presented increased degree centrality in the LING.L, LING.R, and IOG. L and nodal local efficiency in PHG. L, IOG. R, FFG. L, and the HES. L, and decreased betweenness centrality in the PCG. L and CUN. L, degree centrality in the ORBsupmed. R, and nodal local efficiency in ANG. R. Conclusion: Cognitive decline in PSCI patients may be related to BFN disorders;acupuncture therapy may modulate the topological properties of the BFNs of PSCI patients.

Change in Functional Status from Preoperative to One Year Postoperative in Patients Who Have Undergone Elective Open-Heart Surgery: A Repeated-Measures Study

Purpose: Although patient-related factors affect surgical outcomes, preoperative functional status is not measured by any cardiac risk score. Functional status can, however, be objectively measured using validated outcome tools such as the Late-Life Function and Disability Instrument (LLFDI). The purpose of this study was to determine 1) if there was a change over time in functional status, as measured by the LLFDI, in patients who underwent elective cardiac surgery, and if so, 2) what specific aspect(s) of functional status changed. Methods: A prospective longitudinal study of one year was conducted on elective cardiac surgery patients (n = 43) using the self-reported LLFDI, which measures Disability Frequency (frequency of participation in social tasks), Disability Limitation (ability to participate in social tasks) and Function Total (ease in performing routine activities). Higher scores indicate increased function and decreased disability. LLFDI scores were compared at three times (preoperative, six-week and one-year postoperative) using repeated measures ANOVA. Post hoc pairwise comparison was conducted for specific interactions. Results: Both Function Total and Disability Frequency significantly changed over time (p = 0.047 and p = 0.013, respectively). Specifically, patients’ function level was significantly higher one-year postoperative compared to preoperative (M difference = +3.48, SE = 1.48, p = 0.026). Likewise, Disability Frequency scores were significantly higher (i.e. more active) at one-year postoperative versus preoperative (M difference= +5.98, SE = 2.19, p = 0.033). Disability Limitation scores were not significantly different between any time points (p > 0.05). Conclusion: By one-year postoperative, patients demonstrated increased ease in their routine physical activities and were more participatory in social life tasks. Individuals who underwent elective cardiac surgery took more than six weeks to detect notable improvement in functional status, which was expected with a sternotomy approach. This study provides support for the use of the LLFDI as an effective tool to capture functional status in the cardiac population. These findings may assist cardiac patients in recovery timeline expectations.

Proof of Riemann Conjecture Based on Contradiction between Xi-Function and Its Product Expression

Riemann proved three results: analytically continue ζ(s) over the whole complex plane s =σ + it with a pole s =1;(Theorem A) functional equation ξ(t) = G(s0)ζ (s0), s0 =1/2 + it and (Theorem B) product expression ξ1(t) by all roots of ξ(t). He stated Riemann conjecture (RC): All roots of ξ (t) are real. We find a mistake of Riemann: he used the same notation ξ(t) in two theorems. Theorem B must contain complex roots;it conflicts with RC. Thus theorem B can only be used by contradiction. Our research can be completed on s0 =1/2 + it. Using all real roots rk and (true) complex roots zj = tj + iaj of ξ (z), define product expressions w(t), w(0) =ξ(0) and Q(t) > 0, Q(0) =1 respectively, so ξ1(t) = w(t)Q(t). Define infinite point-set L(ω) = {t : t ≥10 and |ζ(s0)| =ω} for small ω > 0. If ξ(t) has complex roots, then ω =ωQ(t) on L(ω). Finally in a large interval of the first module |z1|>>1, we can find many points t ∈ L(ω) to make Q(t) . This contraction proves RC. In addition, Riemann hypothesis (RH) ζ for also holds, but it cannot be proved by ζ.

Status of Renal and Liver Function in Rheumatoid Arthritis (RA) Patients of Chattogram, Bangladesh Treated with Methotrexate (MTX)

Rheumatoid Arthritis (RA) is a chronic autoimmune disorder that is usually manifested as inflammation in multiple joints and several extra-articular symptoms, involving the liver, kidney, eye, skin, blood, blood vessels, heart, lungs, nervous system, and other organs. Methotrexate (MTX) is the anchor drug that treats RA. As renal and liver abnormalities are more common during disease conditions as well as during the treatment period, we tried to find out if there is any impact of MTX in these organs during the treatment of RA patients. Once the disease complications are developed, it is quite difficult to reverse the disease, and treatment in this situation is not very effective. Consequently, patients suffer a lot. So, early evaluation of renal and liver function is essential for the treatment of RA patients and it might also help prevent different complications which are usually very frequently observed. This was a cross-sectional study. A total of 150 RA patients treated with MTX were evaluated for the study where female and male respondents were 115 and 35 respectively. In this study, we found that 82% of RA patients had creatinine levels ≤ 1.1 mg/dL although the normal range of serum creatinine is below 1.4 mg/dL. Usually, a 15% increase in Serum creatinine level from the baseline is considered renal impairment. We found 4% of such cases. Moreover, 2% of RA patients had creatinine levels above the normal range of 1.4 mg/dL and those patients were hypertensive as well. So, a total (4 + 2 = 6)% had renal impairments. Among them, 5% had diabetes mellitus. On the other hand, the ultrasonogram (USG) of RA patients with kidney disease showed signs of renal parenchymal disease and 3% of RA patients having renal problems whose serum creatinine level was within the normal range showed signs of chronic kidney disease (CKD). On the other hand, 2% of RA patients showed signs of hepatic parenchymal disease. In this study, 69% of RA patients had ALT levels ≤ 50 mg/dL, 23% had 50 - 100 mg/dL, and 5% had 101 - 150 mg/dL. The remaining 3% of RA patients had ALT levels above 150 mg/dL. All those patients with ALT levels above 100 mg/dL used Nonsteroidal anti-inflammatory drugs (NSAIDs) concomitantly. Different parameters of liver and renal function should be monitored strongly in RA patients treated with MTX and NSAIDs. MTX should not be given for a prolonged period without monitoring renal and liver function. As MTX, Diabetes Mellitus, Hypertension, etc., may cause renal complications, we could not concretely conclude which one is the actual causative agent.

A New Approach for Solving Fuzzy Linear Multi-Criterion Problems: An Approach Based on Minimization of the Errors Functions

The main purpose of this paper is to build a new approach for solving a fuzzy linear multi-criterion problem by defining a function called “error function”. For this end, the concept of level set is used to construct the error function. In addition, we introduce the concept of deviation variable in the definition of the error function. The algorithm of the new approach is summarized in three main steps: first, we transform the original fuzzy problem into a deterministic one by choosing a specific level . second, we solve separately each uni-criteria problem and we compute the error function for each criteria. Finally, we minimize the sum of error functions in order to obtain the desired compromise solution. A numerical example is done for a comparative study with some existing approaches to show the effectiveness of the new approach.

Stability Analysis of a Self-Memory Prey-Predator Diffusion Model Based on Bazykin Functional Response

To investigate the effects of self-memory diffusion on predator-prey models, we consider a predator-prey model with Bazykin functional response of self- memory diffusion. The uniqueness, boundedness, positivity, existence and stability of equilibrium point of the model are studied. In this paper, the uniqueness of the solution is discussed under the non-negative initial function and Neumann boundary conditions satisfying a specific space. The boundness of the solution is proved by the comparison principle of parabolic equations, and the positivity of the solution is proved by the strong maximum principle of parabolic equations. Hurwitz criterion and Lyapunov function construction are used to analyze the local stability and global stability of feasible equilibrium points. The results show that the system solution is unique non-negative and bounded. The model is unstable at the trivial equilibrium point E0 and the boundary equilibrium point E1, and the condition of whether the positive equilibrium point E2 is stable under certain conditions is given.

Efficacy of a Kiwifruit Extract (PhenActivTM) on Gastrointestinal Tract Function: A Randomised Double-Blind Placebo-Controlled Study

Objective: Gastrointestinal (GI) discomfort is experienced by millions of people every day. This study aimed to evaluate the effect of PhenActivTM, a novel green kiwifruit extract, on gastrointestinal tract (GIT) function in otherwise healthy adults. Methods: 41 healthy adults with mild GI discomfort were enrolled in this double-blind, randomized, placebo-controlled study. Participants were randomized to either take 3.0 g/day of PhenActivTM or a placebo for 6 weeks. Interviews were conducted at baseline, week 3 and week 6, with participants completing questionnaires regarding GI symptoms. Frequency of bowel movements was self-recorded daily. Results: There were no differences in daily and weekly defecation frequency and stool characteristics in either group. The active and placebo groups significantly improve GSRS scores (p , only the active group had a significant improvement in the IBSSS and PAC-QOL scores (p < 0.05) from baseline. Neither group had changes in sleep quality, quality of life and fatigue, plasma zonulin concentrations or macular pigment optical density scores. The product was well tolerated with no GI disturbances or adverse events being reported. Conclusion: Supplementation of 3.0 g/day of PhenActivTM for 6 weeks did not improve defecation frequency or stool composition in healthy adults, but did improve perceived symptoms of GIT function, including symptoms of functional GIT disorders, IBS and constipation. The product was well tolerated and future trials investigating higher doses with more participants and/or a different population would be beneficial.

Unsupervised Functional Data Clustering Based on Adaptive Weights

In recent years, functional data has been widely used in finance, medicine, biology and other fields. The current clustering analysis can solve the problems in finite-dimensional space, but it is difficult to be directly used for the clustering of functional data. In this paper, we propose a new unsupervised clustering algorithm based on adaptive weights. In the absence of initialization parameter, we use entropy-type penalty terms and fuzzy partition matrix to find the optimal number of clusters. At the same time, we introduce a measure based on adaptive weights to reflect the difference in information content between different clustering metrics. Simulation experiments show that the proposed algorithm has higher purity than some algorithms.

The Clinical Association of Left Atrial Function with Left Ventricular Ejection Fraction

The left atrium (LA) has been recognized as a morphophysiological barometer of left ventricular (LV) diastolic dysfunction. Because in the myocardial ischemia cascade where LV diastolic dysfunction often precedes LV systolic dysfunction, the LA which fashions as an early marker of diastolic anomaly, could equally reflect a declining LV function and/or be a good predictor of potential sequelae. We assessed this association of LA function with reduced LV systolic function among hospitalized patients. Among patients with reduced LV ejection fraction, LA passive ejection fraction was lower (0.172 ± 0.12 vs. 0.232 ± 0.14, p = 0.013) whereas LA kinetic energy was higher (6.48 ± 6.3 vs. 4.57 ± 3.5, p = 0.005). Echocardiographic assessment of LA function, therefore, appears correlated with LVEF and could be important when risk stratifying hospitalized patients.

A Systemic Functional Interpretation of English Expletives at the Ideational and Interpersonal Level

This thesis put the expletives into the whole language system to analyze, using the frame of the Halliday's(1985) System-Functional Linguistics Theory. The expletives can be analyzed on the sentential level, including the ideational function and the interpersonal function.

Textual Function of Expletives in Godfather

This thesisanalyzes the expletives in the film Godfather with the theoretical frame of Halliday's(1985) System-Functional Linguistics Theory,especially from the perspective of the textual function of expletives,namely,we will elaborate the textual function with the film Godfather as the example. The expletives may be very appropriate and necessary in certain situation since they have a cohesive function.

Sub-Differential Characterizations of Non-Smooth Lower Semi-Continuous Pseudo-Convex Functions on Real Banach Spaces

In this paper, we characterize lower semi-continuous pseudo-convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the pseudo-monotonicity of its Clarke-Rockafellar Sub-differential. We extend the results on the characterizations of non-smooth convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the monotonicity of its sub-differentials to the lower semi-continuous pseudo-convex functions on real Banach spaces.

Literary Text Translation from Functional Perspective

Functional linguistics has become the most influential linguistic theory in the field of translation study since the last de cade of 20th century.However,for the complexity of the context,it is uncommon and insufficient to explore literary texts.Draw ing a general picture of literary text translation from functional perspectives aims to probe into the functional modes of translation.

Duality between Bessel Functions and Chebyshev Polynomials in Expansions of Functions

In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.

The Application of Systemic-Functional Linguistics in Discourse Analysis

Systemic-functional linguistics considers function and semantics as the basis of human language and communicative activity. In recent years, Systemic-functional linguistics has been applied in such areas as natural language processing, early language learning, language education, stylistics, foreign language teaching, translation and stylistics. It sheds new light on the further application of this linguistic theory in other fields.

Functions of Discourse Markers

Discourse markers are pervasive in conversation, generally used to facilitate human communication. With respect to different contextual effects that discourse markers have in utterance interpretation, they may function subjectively, interactionally or textually.