Finiteness of Mapping Degree Sets for 3-Manifolds

By constructing certain maps, this note completes the answer of the question： For which closed orientable 3-manifold N, is the set of mapping degrees D（M, N） finite for any closed orientable 3-manifold M？

On Self-Mapping Degrees of S^3-Geometry Manifolds

In this paper we determined all of the possible self-mapping degrees of the manifolds with S3-geometry, which are supposed to be all 3-manifolds with finite fundamental groups. This is a part of a project to determine all possible self-mapping degrees of all closed orientable 3-manifold in Thurston＇s picture.

Comparison of detection results of hypoxic-ischemic encephalopathy at different degrees in infant patients between brain electrical activity mapping, transcranial Doppler sonography and computer tomography examinations

BACKGROUND： It has been proved that brain electrical activity mapping （BEAM） and transcranial Doppler （TCD） detection can reflect the function of brain cell and its diseased degree of infant patients with moderate to severe hypoxic-ischemic encephalopathy （HIE）. OBJECTIVE： To observe the abnormal results of HIE at different degrees detected with BEAM and TCD in infant patients, and compare the detection results at the same time point between BEAM, TCD and computer tomography （CT） examinations. DESIGN ： Contrast observation SETTING： Departments of Neuro-electrophysiology and Pediatrics, Second Affiliated Hospital of Qiqihar Medical College. PARTICIPANTS： Totally 416 infant patients with HIE who received treatment in the Department of Newborn Infants, Second Affiliated Hospital of Qiqihar Medical College during January 2001 and December 2005. The infant patients, 278 male and 138 female, were at embryonic 37 to 42 weeks and weighing 2.0 to 4.1 kg, and they were diagnosed with CT and met the diagnostic criteria of HIE of newborn infants compiled by Department of Neonatology, Pediatric Academy, Chinese Medical Association. According to diagnostic criteria, 130 patients were mild abnormal, 196 moderate abnormal and 90 severe abnormal. The relatives of all the infant patients were informed of the experiment. METHOOS： BEAM and TCD examinations were performed in the involved 416 infant patients with HIE at different degrees with DYD2000 16-channel BEAM instrument and EME-2000 ultrasonograph before preliminary diagnosis treatment （within 1 month after birth） and 1,3,6,12 and 24 months after birth, and detected results were compared between BEAM, TCD and CT examinations. MAIN OUTCOME MEASURES： Comparison of detection results of HIE at different time points in infant patients between BEAM. TCD and CT examinations. RESULTS： All the 416 infant patients with HIE participated in the result analysis. （1） Comparison of the detected results in infant patients with mild HIE at different time points after birth between BEAM, TCD and CT examinations： BEAM examination showed that the recovery was delayed, and the abnormal rate of BEAM examination was significantly higher than that of CT examination 1 and 3 months after birth [55.4%（72/130）vs. 17.0% （22/130 ）,x^2=41.66 ;29.2% （ 38/130 ） vs. 6.2% （ 8/130 ）, x^2=23.77, P 〈 0.01 ], exceptional patients had mild abnormality and reached the normal level in about 6 months. TCD examination showed that the disease condition significantly improved and infant patients with HIE basically recovered 1 or 2 months after birth, while CT examination showed that infant patients recovered 3 or 4 months after birth. （2） Comparison of detection results of infant patients with moderate HIE at different time points between BEAM, TCD and CT examinations： The abnormal rate of BEAM examination was significantly higher than that of CT examination 1,3,6 and 12 months after birth [90.8% （178/196）,78.6% （154/196）,x^2=4.32,P 〈 0.05;64.3% （126/196）,43.9% （86/196） ,x^2=16.44 ;44.9% （88/196） ,22.4% （44/196）,x^2=22.11 ;21.4% （42/196）, 10.2% （20/196）,x^2=9.27, P 〈 0.01]. BEAM examination showed that there was still one patient who did not completely recovered in the 24^th month due to the relatives of infant patients did not combine the treatment,. TCD examination showed that the abnormal rate was 23.1%（30/196）in the 1^st month after birth, and all the patients recovered to the normal in the 3^rd month after birth, while CT examination showed that mild abnormality still existed in the 24^th month after birth （1.0% ,2/196）. （3） Comparison of detection results of infant patients with severe HIE at different time points between BEAM, TCD and CT examinations： The abnormal rate of BEAM examination was significantly higher than that of CT examination in the 1^st, 3^rd, 6^th and 12^th months after birth[86.7% （78/90）,44.4% （40/90）,x^2=35.53;62.2% （56/90）,31.1% （28/90）,x^2=17.51 ;37.8% （34/90）,6.7% （6/90）, x^2=27.14, P 〈 0.01]. BEAM examination showed that mild abnormality still existed in 4 infant patients in the 24^th month after birth. TCD examination showed that the abnormal rate was 11.1% （10/90） in the 3^rd month after birth, and all the infant patients recovered in the 6^th month after birth. CT examination showed that the abnormal rate was 6.7%（6/90） in the 12^th month after birth, and all of infant patients recovered to the normal in the 24^th month after birth.CONCLUSION ： BEAM is the direct index to detect brain function of infant patients with HIE, and positive reaction is still very sensitive in the tracking detection of convalescent period. The positive rate of morphological reaction in CT examination is superior to that in TCD examination, and the positive rate is very high in the acute period of HIE in examination.

The degree of biholomorphic mappings between special domains in C^n preserving 0

Let G_i be a closed Lie subgroup of U(n), ?_i be a bounded Gi-invariant domain in C^n which contains 0, and O(C^n)^(Gi) = C, for i = 1, 2. If f : ?_1→ ?_2 is a biholomorphism, and f(0) = 0, then f is a polynomial mapping(see Ning et al.(2017)). In this paper, we provide an upper bound for the degree of such polynomial mappings. It is a natural generalization of the well-known Cartan's theorem.

The Degree of Proper Holomorphic MappingsBetween Special Domains in C^n

Let Gi be closed Lie groups of U (n), Ω i be bounded Gi-invariant domains in C^n which contains 0, and O(C^n)^Gi = C, for i = 1, 2. It is known that if f : Ω 1 → Ω 2 is a proper holomorphic mapping, and f^-1{0} = {0}, then f is a polynomial mapping. In this paper, we provide an upper bound for the degree of such a polynomial mapping using the multiplicity of f .

Non-Zero Degree Maps Between 2n-Manifolds

Thom–Pontrjagin constructions are used to give a computable necessary and sufficient condition for a homomorphism ? : H n (L;Z) → H n (M;Z) to be realized by a map f : M → L of degree k for closed (n ? 1)-connected 2n-manifolds M and L, n > 1. A corollary is that each (n ? 1)-connected 2n-manifold admits selfmaps of degree larger than 1, n > 1. In the most interesting case of dimension 4, with the additional surgery arguments we give a necessary and sufficient condition for the existence of a degree k map from a closed orientable 4-manifold M to a closed simply connected 4-manifold L in terms of their intersection forms; in particular, there is a map f : M → L of degree 1 if and only if the intersection form of L is isomorphic to a direct summand of that of M.

Realization of Numbers As the Degrees of Maps between Manifolds

We determine the set of degrees between some classes of oriented closed （n - 1）-connected 2n-manifolds by using the arithmetic theory of quadratic forms.

The Fuzzy Incidence Degree in Systems Analysis

In this paper, the computational formula of fuzzy incidence degree in systems analysis is given. Besides, we discuss ifs simple properties. and give a computation example at last.

A Partial Order in the Knot Table Ⅱ

A partial order on the set of the prime knots can be defined by the existence of a surjective homomorphism between knot groups. In the previous paper, we determined the partial order in the knot table. In this paper, we prove that 31 and 41 are minimal elements. Further, we study which surjection a pair of a periodic knot and its quotient knot induces, and which surjection a degree one map can induce.

Generalized Hopfian Property, a Minimal Haken Manifold, and Epimorphisms Between 3-Manifold Groups

We address the question that if π_1-surjective maps between closed aspherical 3-manifolds have the same rank on π_1 they must be of non-zero degree. The positive answer is proved for Seifert manifolds, which is used in constructing the first known example of minimal Haken manifold. Another motivation is to study epimorphisms of 3-manifold groups via maps of non-zero degree between 3- manifolds. Many examples are given.