Virtual Reality-Based Mirror Therapy for Upper Extremity Function among Stroke Patients:A Meta-analysis and Systematic Review

[Objectives]To investigate the evidence-based effect of virtual reality-based mirror therapy system(VR-MT)on upper extremity function among stroke patients.[Methods]A systematic electronic searching of the Medline,PubMed,Web of Science and CNKI was initially performed up to June 10,2024.The risk of bias of the included studies was evaluated using RevMan 5.4 software based on the Cochrane Handbook for Systematic Reviews.The random-effects model or fixed-effects models was employed to estimate the standardized mean difference(SMD).The subgroup analyses were conducted exploring theVR-MT type(immersive or non-immersive)and comparing with MT or control group.[Results]In total 8 studies with a total of 273 stroke patients were included in this review.The pooled analysis of these trials showed a statistically significant enhancement inFMA-UE scores(6 studies,SMD=0.72,[95%CI 0.37 to 1.06];P<0.0001,I^(2)=31%)and Box and Block Test(BBT)(3 studies,SMD=0.49,[95%C/0.05 to 0.93];P=0.03,I^(2)=0%),rather than Manual Function Test(MFT)scores(3 studies,SMD=0.38,[95%CI-0.09 to 0.84];P=0.11,I^(2)=0%)following the application of reality-based mirror therapy.Additionally,the subgroup analysis results indicated that immersive VR-MT can significantly improve FMA-UE(5studies,SMD=0.73,[95%CI 0.24 to 1.23];P=0.004,I^(2)=43%).In contrast,the overall effect of non-immersive VR-MT was non-significant(2 studies,SMD=0.33,[95%CI-0.69 to 1.34];P=0.53,I^(2)=72%).[Conclusions]In this systematic review and meta-analysis,our findings indicate that immersiveVR-MT has the potential to improve upper extremity function among stroke patients.

Effects of Repetitive TranscranialMagnetic Stimulation on Upper Extremity Function in Acute Stroke:A Meta-analysis and Systematic Review

[Objectives]To conduct a comprehensive examination of the evidence-based impact of repetitive transcranial magnetic stimulation(rTMS)on upper extremity functionality in patients with acute stroke.[Methods]A rigorous and systematic electronic search was conducted across the Medline,PubMed,and Web of Science databases,encompassing literature up to July 1,2024.To ensure the reliability of the in-cluded studies,an assessment of their risk of bias was conducted using RevMan 5.4 software,in accordance with the rigorous standards out-lined in the Cochrane Handbook for Systematic Revieus.Subsequently,we employed either the random-effects model or the fixed-effects model,depending on the heterogeneity of the data,to estimate the standardized mean difference(SMD)in outcomes,utilizing Stata 18.0 software for statistical analysis.[Results]Our review encompassed a total of five studies,involving 252 patients with acute stroke.The pooled analysis of these studies revealed a statistically significant improvement in Fugl-Meyer Assessment of the Upper Extremity(FMA-UE)scores among pa-tients who received rTMS therapy(SMD=2.71,95%CI:0.85 to 4.56;P<0.0001),albeit with considerable heterogeneity(I^(2)=97.65%)across the trials.[Conclusions]The results of this systematic review and meta-analysis underscore the promising potential of rTMS in enhancing upper extremity function in patients who have experienced an acute stroke.These findings provide compelling evidence for the therapeutic benefits of rTMS in this patient population.

Motor relearning program and Bobath method improve motor function of the upper extremities in patients with stroke

BACKGROUND: In the natural evolution of cerebrovascular disease, unconscious use of affected extremity during drug treatment and daily life can improve the function of affected upper extremity partially, but it is very slow and also accompanied by the formation of abnormal mode. Therefore, functional training should be emphasized in recovering the motor function of extremity. OBJECTIVE: To observe the effects of combination of motor relearning program and Bobath method on motor function of upper extremity of patients with stroke. DESIGN: Comparison of therapeutic effects taking stroke patients as observation subjects. SETTING: Department of Neurology, General Hospital of Beijing Jingmei Group. PARTICIPANTS: Totally 120 stroke patients, including 60 males and 60 females, averaged (59±3) years, who hospitalized in the Department of Neurology, General Hospital of Beijing Jingmei Group between January 2005 and June 2006 were recruited. The involved patients met the following criteria: Stroke attack within 2 weeks; diagnosis criteria of cerebral hemorrhage or infarction made in the 4th National Cerebrovascular Disease Conference; confirmed by skull CT or MRI; Informed consents of therapeutic regimen were obtained. The patients were assigned into 2 groups according to their wills: rehabilitation group and control group, with 30 males and 30 females in each group. Patients in rehabilitation group averaged (59±2)years old, and those in the control group averaged (58±2)years old. METHODS: ① Patients in two groups received routine treatment in the Department of Neurology. When the vital signs of patients in the rehabilitation group were stable, individualized treatment was conducted by combined application of motor relearning program and Bobath method. Meanwhile, training of activity of daily living was performed according to the disease condition changes of patients at different phases, including the nursing and instruction of body posture, the maintenance of good extremity position, bed exercise, bedside sit up and sitting position balance, sit up exercise, dynamic and static balance exercise, walking exercise, active training and passive training. The strength, time and speed of training were increased gradually according to their physical abilities. Patients were trained 45 to 60 minutes once, 5 times a week, within 2 weeks. ② Evaluation criteria of therapeutic effect: The motor function of upper extremity was evaluated by Fugl-Meyer method on the day of beginning and end of treatment. Higher points indicated better function of upper extremity. ③ t test and paired t test were used for comparing the difference of intergroup and intragroup measurement data, respectively. MAIN OUTCOME MEASURES: Changes in Fugl-Meyer scoring of two groups before and after treatment. RESULTS: Totally 120 stroke patients participated in the final analysis. Before treatment, Fugl-Meyer scoring was close between rehabilitation group and control group [(14.47±2.38),(14.16±2.39) points, P > 0.05]; Fugl-Meyer scoring of rehabilitation group after treatment was significantly higher than that before treatment and that of control group[(37.93±2.67),(18.36±2.43) points, t =11.053, 5.408, P < 0.01]; There were no significant differences in Fugl-Meyer scoring between before treatment in the control group and control group (P > 0.05). CONCLUSION: Combined application of motor relearning program and Bobath method can significantly improve the motor function of upper extremity of patients with stroke.

ON APPROXIMATION AND GENERALIZED TYPE OF ANALYTIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES

In the present paper, we study the polynomial approximation of analytic functions of several complex variables. The characterizations of generalized type of analytic functions of several complex variables have been obtained in terms of approximation and interpolation errors.

Functional magnetic resonance imaging evidence for activated functional brain areas following acupoint needling in the extremities

Totally three articles focusing on functional magnetic resonance imaging features of brain function in the activated brain regions of stroke patients undergoing acupuncture on the healthy limbs and healthy controls undergoing acupuncture on the lower extremities are published in three issues. We hope that our readers find these papers useful to their research.

Constraint-induced movement therapy in treatment of acute and sub-acute stroke: a meta-analysis of 16 randomized controlled trials

OBJECTIVE： The aim of this meta-analysis was to evaluate the clinical efficacy of constraint-induced movement therapy in acute and sub-acute stroke. DATA SOURCES： The key words were stroke, cerebrovascular accident, constraint-induced therapy, forced use, and randomized controlled trial. The databases, including China National Knowledge Infrastructure, WanFang, Weipu Information Resources System, Chinese Biomedical Literature Database, PubMed, Med- line, Embase, the Cochrane Central Register of Controlled Trials, and the Cochrane Database of Systematic Reviews, were searched for studies on randomized controlled trials for treating acute or sub-acute stroke published before March 2016. DATA SELECTION： We retrieved relevant randomized controlled trials that compared constraint-induced movement therapy in treatment of acute or sub-acute stroke with traditional rehabilitation therapy （tradi- tional occupational therapy）. Patients were older than 18 years, had disease courses less than 6 months, and were evaluated with at least one upper extremity function scale. Study quality was evaluated, and data that met the criteria were extracted. Stata 11.0 software was used for the meta-analysis. OUTCOME MEASURES： Fugl-Meyer motor assessment of the arm, the action research-arm test, a motor activity log for amount of use and quality of movement, the Wolf motor function test, and a modified Bar- thel index. RESULTS： A total of 16 prospective randomized controlled trials （379 patients in the constraint-induced movement-therapy group and 359 in the control group） met inclusion criteria. Analysis showed significant mean differences in favor of constraint-induced movement therapy for the Fugl-Meyer motor assessment of the arm （weighted mean difference （WMD） = 10.822; 95% confidence intervals （95% CI）： 7.419-14.226）, the action research-arm test （WMD = 10.718; 95% CI： 5.704-15.733）, the motor activity log for amount of use and quality of movement （WMD = 0.812; 95% CI： 0.331-1.293） and the modified Barthel index （WMD = 10.706; 95% CI： 4.417-16.966）. CONCLUSION： Constraint-induced movement therapy may be more beneficial than traditional rehabili- tation therapy for improving upper limb function after acute or sub-acute stroke.

BEST APPROXIMATION FOR WEIERSTRASS TRANSFORM CONNECTED WITH SPHERICAL MEAN OPERATOR

Using reproducing kernels for Hilbert spaces, we give best approximation for Weierstrass transform associated with spherical mean operator. Also, estimates of extremal functions are checked.

Compactness of Extremals for Trudinger-Moser Functionals on the Unit Ball in R^(2)

Let B be a unit ball in R^(2),W_(0)^(1,2)(B)be the standard Sobolev space.For anyϵ>0,de Figueiredo,doÓ,dos Santons,Yang and Zhu proved the existence of extremals of a Trudinger-Moser inequality in the unit ball.Precisely,u∈W_(0)^(1,2)(B)^(sup),∫_(B)|▽u|^(2)dx≤1∫_(B)^(|x|^(2∈)e^(4π(1+∈)u^(2)dx)can be attained by some radially symmetric function u_(∈)∈W_(0)^(1,2)(B)∫_(B)with∫_(B)|▽u|^(2)dx=1.In this note,we concern the compactness of the function family{uϵ}ϵ>0 and prove that up to a subsequence uϵconverges to some function u 0 in C^(1)(B)asϵ→0.Furthermore,u_(0) is an extremal function of the supremum u∈W_(0)^(1,2)(B)^(sup),∫_(B)|▽u|^(2)dx≤1∫_(B)e^(4πu^(2)dx)Let us explain the result in geometry.Denoteω_(0)=dx_(1)^(2)+dx_(2)^(2)be the standard Euclidean metricω_(∈)|x|^(2ϵ)ω_(0)for xϵB..Then the extremal family{u_(∈)}∈>0 of the following Trudinger-Moser functionals∫_(B)e^(4π(1+∈)u^(2))dvω_(∈)under the constraint W_(0)^(1,2)(B)and∫_(B)|▽ω_∈)u|^(2)dvω_∈)≤1 is compact asϵ→0.

Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds

Let (M,g) be a compact Riemannian manifold without boundary, and (N,g) a compact Riemannian manifold with boundary. We will prove in this paper that the and can be attained. Our proof uses the blow-up analysis.

Extremal Functions of the Singular Moser-Trudinger Inequality Involving the Eigenvalue

In this paper, we derive the singular Moser-Trudinger inequality which in-volves the first eigenvalue and several singular points, and further prove the existenceof the extremal functions for the relative Moser-Trudinger functional. Since the prob-lems involve more complicated norm and multiple singular points, not only we can＇tuse the symmetrization to deal with a one-dimensional inequality, but also the pro-cesses of the blow-up analysis become more delicate. In particular, the new inequalityis more general than that of [1, 2].

Remarks on the Extremal Functions for the Moser-Trudinger Inequality

We will show in this paper that if A is very close to 1, thenI（M,λ,m） =supu∈H0^1,n（m）,∫m｜△↓u｜^ndV=1∫Ω（e^αn｜u｜^n/（n-1）-λm∑k=1｜αnun/（n-1）｜k/k!）dVcan be attained, where M is a compact-manifold with boundary. This result gives a counter-example to the conjecture of de Figueiredo and Ruf in their paper titled ＂On an inequality by Trudinger and Moser and related elliptic equations＂ （Comm. Pure. Appl. Math., 55, 135-152, 2002）.

Estimation of maximum inclusion by statistics of extreme values method in bearing steel

A statistic method, statistics of extreme values （SEV）, was described in detail, which can esti mate the size of maximum inclusion in steel. The characteristic size of the maximum inclusion in a high clean bearing steel （GCrl5） was evaluated by this method, and the morphology and corn position of large inclusions found were analyzed by scanning electron microscopy （SEM）. When standard inspection area （S0） is 280 mm2, the characteristic size of the biggest inclusion found in 30 standard inspection area is 23.93 μm, and it has a 99.9% probability of the characteristic size of maximum inclusion predicted being no larger than 36.85μm in the experimental steel. SEM result shows that large inclusions found are mainly composed of CaS, calcium-aluminate and MgO. Compositing widely exists in large inclusions in high clean bearing steel. Compared with traditional evaluation method, SEV method mainly focuses on inclusion size, and the esti- mation result is not affected by inclusion types. SEV method is suitable for the inclusion eval uation of high clean bearing steel.

Construction of Optimal Quadrature Formulas Exact for Exponentional-trigonometric Functions by Sobolev's Method

The paper studies Sard's problem on construction of optimal quadrature formulas in the space W_(2)^((m,0))by Sobolev's method.This problem consists of two parts:first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas.Here the norm of the error functional is calculated with the help of the extremal function.Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space W_(2)^((m,0)) is obtained,moreover the existence and uniqueness of the solution of this system are discussed.Next,the discrete analogue D_(m)(hβ)of the differential operatord^(2m)/dx^(2m)-1 is constructed.Further,Sobolev's method of construction of optimal quadrature formulas in the space W_(2)^((m,0)),which based on the discrete analogue D_(m)(hβ),is described.Next,for m=1 and m=3 the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained.Finally,at the end of the paper the rate of convergence of the optimal quadrature formulas in the space W_(2)^((3,0))for the cases m=1 and m=3 are presented.

Extremal Functions for Adams Inequalities in Dimension Four

Let Ω (∈) R4 be a smooth bounded domain,W2,20(Ω) be the usual Sobolev space.For any positive integer e,λl(Ω) is the e-th eigenvalue of the bi-Laplacian operator.Define E(l)=Eλ1(Ω) ⊕ Eλ2(Ω)(⊕) … ⊕Eλe(Ω),where Eλi(Ω) is eigenfunction space associated with λi(Ω).E⊥e denotes the orthogonal complement of Ee in W2,20 (Ω).For 0 ≤ α ≤ λe+1 (Ω),we define a norm by ||u||22,α =||△u||22-α||u||22 for u ∈ E⊥e.In this paper,using the blow-up analysis,we prove the following Adams inequalities supu∈E⊥e,||u||2,α≤1∫Ω e32π2u2 dx < +∞;moreover,the above supremum can be attained by a function u0 ∈ E⊥e∩C4((Ω)) with || u0||2,α =1.This result extends that of Yang (J.Differential Equations,2015),and complements that of Lu and Yang (Adv.Math.2009) and Nguyen (arXiv: 1701.08249,2017).

Extremal Problems of Bloch Function Spaces

Let f be analytic in a hyperbolic region Ω. The Bloch constant β_f of f is defined by β_f=sup z∈Ω|f’(z)|/λ_Ω(z), where λ_Ω(z)|dz|is the Poincare metric in Ω. Suppose △is hyperbolic and lim inf ω→cλ△(w)】λ(△)】0, (?)c∈(?)△ where λ(△)=inf w∈△ λ△(w). Then for all f with f(Ω)(?)△, we have β_f≤1/λ(△). In this paper we study the extremal functions defined by β_f=1/λ(△) and the existence of those functions.

Extremal Functions for an Improved Trudinger-Moser Inequality Involving L^(P)-Norm in R^(n)

Let W^(1,n)(R^(n))be the standard Sobolev space.For any T>0 and p>n>2,we denote■Define a norm in W^(1,n)(R^(n))by■where 0≤α<λ_(n,p).Using a rearrangement argument and blow-up analysis,we will prove■can be attained by some function u_(0)∈W^(1,n)(R^(n))∩C^(1)(R^(n))with ||u_(0)||_(n,p)=1,here a_(n)=n■_(n-1)^(1/n-1) and ■_(n-1) is the measure of the unit sphere in R^(n).

Extremal Functions for Trudinger-Moser Type Inequalities in RN

Let N≥2, aN=Nω1 N-1（N-1） , where ωN-1 denotes the area of the unit sphere in RN. In this note, we prove that for any 0 〈a〈aN and any β 〉 0, the supremum sup u∈W 1,N（R N）,｜｜u｜｜W 1,N（R N）≤1 ∫ R N ｜u｜β （e a ｜u｜ N/N-1 N-2 ∑j=0 aj/j!｜u｜ Ni/N-1）dx.can be attained by some function u ∈ W1, N （R N） with ｜｜ u ｜｜ W 1,N （R N ） = 1. Moreover, when a ≥ aN, the above supremum is infinity.

A Trudinger-Moser Inequality Involving Lp-norm on a Closed Riemann Surface

In this paper,using the method of blow-up analysis,we obtained a Trudinger–Moser inequality involving L^(p)-norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trudinger–Moser functional.

Finsler Trudinger-Moser inequalities on R^(2)

The first aim of this article is to study the sharp singular(two-weight)Trudinger-Moser inequalities with Finsler norms on R^(2).The second goal is to propose a different approach to study a vanishing-concentration-compactness principle for the Trudinger-Moser inequalities and use this to investigate the existence and the nonexistence of the maximizers for the Trudinger-Moser inequalities in the subcritical regions.Finally,by applying our Finsler Trudinger-Moser inequalities to suitable Finsler norms,we derive the sharp affine Trudinger-Moser inequalities which are essentially stronger than the Trudinger-Moser inequalities with standard energy of the gradient.

Critical Trace Trudinger-Moser Inequalities on a Compact Riemann Surface with Smooth Boundary

In this paper,the author concerns two trace Trudinger-Moser inequalities and obtains the corresponding extremal functions on a compact Riemann surface(Σ,g)with smooth boundaryθΣ.Explicitly,letλ_(1)(θΣ)=inf_(u∈W^(1,2)(Σ,g),∫_(θΣ)uds_(g)=0,u≠0∫_(Σ)(|▽_(g)u|^(2)+u^(2))dv_(g)/∫_(θΣ)u^(2)ds_(g)and H={u∈W^(1,2)(Σ,g):∫_(Σ)(|▽_(g)u|^(2)+u^(2))dv_(g)-α∫_(θΣ)u^(2)ds_(g)≤1 and∫_(θΣ)uds_(g)=0},where W^(1,2)(Σ,g)denotes the usual Sobolev space and▽g stands for the gradient operator.By the method of blow-up analysis,we obtain sup_(u∈H)∫_(θΣ)e^(πu^(2))ds_(g){<+∞,0≤α﹤λ_(1)(∂Σ),=+∞,α≥λ_(1)(∂Σ)Moreover,the author proves the above supremum is attained by a function u∈H∩C^(∞)(∑)for any 0≤α<λ_(1)(θΣ).Further,he extends the result to the case of higher order eigenvalues.The results generalize those of[Li,Y.and Liu,P.,Moser-Trudinger inequality on the boundary of compact Riemannian surface,Math.Z.,250,2005,363–386],[Yang,Y.,Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary,Pacific J.Math.,227,2006,177–200]and[Yang,Y.,Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two,J.Diff.Eq.,258,2015,3161–3193].