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(s<sub>0</sub>)ζ (s<sub>0</sub>), s&...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(s<sub>0</sub>)ζ (s<sub>0</sub>), s<sub>0</sub> =1/2 + it and (Theorem B) product expression ξ<sub>1</sub>(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 s<sub>0</sub> =1/2 + it. Using all real roots r<sub>k</sub><sub> </sub>and (true) complex roots z<sub>j</sub> = t<sub>j</sub> + ia<sub>j</sub> of ξ (z), define product expressions w(t), w(0) =ξ(0) and Q(t) > 0, Q(0) =1 respectively, so ξ<sub>1</sub>(t) = w(t)Q(t). Define infinite point-set L(ω) = {t : t ≥10 and |ζ(s<sub>0</sub>)| =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 |z<sub>1</sub>|>>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.展开更多
IL-2 production and IL-2 receptor (Tac antigen) of the peripheral blood mononuclear cells in 30 patients with aplastic anemic (AA) were studied. We found that mononuclear cells from patients produce spontaneously IL-2...IL-2 production and IL-2 receptor (Tac antigen) of the peripheral blood mononuclear cells in 30 patients with aplastic anemic (AA) were studied. We found that mononuclear cells from patients produce spontaneously IL-2 in the absence of exogenous lee-tin stimulation, the proportion of Tac+ cells in mononuclear cells increased. The release of IL-2 and or Tac antigen expression were elevated in almost every patient with AA. The plasma from patients stimulate mitogen-induced blastogenesis and Tac antigen expression of normal human lymphocytes. Immunological 1 abnormalities of patients with AA possibly might represents secondary response to bone marrow depression.展开更多
Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that A<sup>n</sup> + B<sup>n</sup> = C<sup>n</sup>, in which n is a natural number great...Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that A<sup>n</sup> + B<sup>n</sup> = C<sup>n</sup>, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A<sup>2</sup> + B<sup>2</sup> + C<sup>2</sup> + D<sup>2</sup> + so on =A<sub>n</sub><sup>2 </sup>where all are natural numbers.展开更多
Riemann (1859) had proved four theorems: analytic continuation ζ(s), functional equation ξ(z)=G(s)ζ(s)(s=1/2+iz, z=t−i(σ−1/2)), product expression ξ1(z)and Riemann-Siegel formula Z(z), and proposed Riemann conjec...Riemann (1859) had proved four theorems: analytic continuation ζ(s), functional equation ξ(z)=G(s)ζ(s)(s=1/2+iz, z=t−i(σ−1/2)), product expression ξ1(z)and Riemann-Siegel formula Z(z), and proposed Riemann conjecture (RC): All roots of ξ(z)are real. We have calculated ξand ζ, and found that ξ(z)is alternative oscillation, which intuitively implies RC, and the property of ζ(s)is not good. Therefore Riemann’s direction is correct, but he used the same notation ξ(t)=ξ1(t)to confuse two concepts. So the product expression only can be used in contraction. We find that if ξhas complex roots, then its structure is destroyed, so RC holds. In our proof, using Riemann’s four theorems is sufficient, needn’t cite other results. Hilbert (1900) proposed Riemann hypothesis (RH): The non-trivial roots of ζhave real part 1/2. Of course, RH also holds, but can not be proved directly by ζ(s).展开更多
In this paper, we consider a general quasi-differential expressions t1,t2 Tn, each of order n with complex coefficients and their formal adjoints are t1＋,t2＋- x＋ on [0, b） respectively. We show in the direct sum s...In this paper, we consider a general quasi-differential expressions t1,t2 Tn, each of order n with complex coefficients and their formal adjoints are t1＋,t2＋- x＋ on [0, b） respectively. We show in the direct sum spaces LZ（Ip）, p = 1,2 N of functions defined on each of the separate intervals with the case of one singular end-points and under suitable conditions on the function F that all solutions of the product quasi-integro differential equations are bounded and LZw -bounded on [0,b）.展开更多
This paper will prove Riemann conjecture(RC): All zeros of <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em>ξ</em></span>(<span style="...This paper will prove Riemann conjecture(RC): All zeros of <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em>ξ</em></span>(<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em>τ</em></span></span></span></span>)</span> lie on critical line. Denote <img src="Edit_189dc2b2-73ef-4036-9f06-ecf8a47fe58b.png" width="140" height="16" alt="" />, and <img src="Edit_a8ec55cb-e4c4-4156-ba23-ae01a31d1bc8.png" width="110" height="22" alt="" /> on critical line. We have found two mysteries in Riemann’s paper. <em>The first mystery</em> is the equivalence: <img src="Edit_3c075830-3c6c-4a23-9851-5b7d219e8000.png" width="140" height="21" alt="" /> is uniquely determined by its initial value <span style="white-space:nowrap;"><em>u</em> (<em>t</em>)</span>. <em>The second mystery</em> is Riemamm conjecture 2 (RC2): Using all zeros <span style="white-space:nowrap;"><em>t<sub>j</sub> </em></span>of <em>u</em> (<em>t</em>) can uniquely express <img src="Edit_b15d9c18-b55b-49e3-97a1-d2e03ccb6343.png" width="175" height="23" alt="" />. We find that the proof of RC is hidden in it. Our basic idea as follows. Consider functional equation <img src="Edit_f5295ff4-90b2-4465-851a-cad140b181c8.png" width="305" height="20" alt="" />. It is known that on critical line <img src="Edit_b45bff49-6d09-456b-9d1f-4259c66293d3.png" width="310" height="23" alt="" /> and <img src="Edit_4182ba79-0fcb-4f84-b7e7-c7574406596e.png" width="85" height="26" alt="" />, then we have the upper bound of growth <img src="Edit_d3d84d75-cc56-47b8-a9a7-ef8a9a5f07b1.png" width="250" height="33" alt="" /> To prove RC2 (or RC), by contradiction. If <span style="white-space:nowrap;"><em>ξ</em>(<em>τ</em>)</span> has conjugate complex roots <em>t</em>'<span style="white-space:nowrap;">±<em>i</em><span style="white-space:nowrap;"><em>β</em></span>'’</span>, <span style="white-space:nowrap;"><em>β</em>'>0</span>, <em>R</em><sup>2</sup>=t'<sup>2</sup>+<span style="white-space:nowrap;"><em>β</em>'<sup>2</sup></span>, by symmetry <span style="white-space:nowrap;"><em>ξ</em>(<em>τ</em>)=<span style="white-space:nowrap;"><em>ξ</em>(-<em>τ</em>)</span></span>, then -(<em>t</em>'<span style="white-space:nowrap;">±<em>i</em><em>β</em>''</span>) do yet. So <em>ξ</em> must contain four factors. Then <em>u</em>(<em>t</em>) contains a real factor <img src="Edit_ac03c1a5-0480-4efa-aac4-7788852a42a9.png" width="225" height="22" alt="" /> and <span style="white-space:nowrap;">ln|<em>u</em>(<em>t</em>)|</span> contains a term (the lower bound) <img src="Edit_6e94ad71-a310-4717-99ee-90384b0d89ba.png" width="230" height="19" alt="" /> which contradicts to the growth above. So <span style="white-space:nowrap;"><em>ξ</em></span> can not have the complex roots and <em>u</em>(<em>t</em>) does not have the factor <em>p</em>(<em>t</em>). Therefore both RC2 and RC are proved. We have seen that the two-dimensional problem is reduced to one-dimension and the one-dimensional <span style="white-space:nowrap;"><em>u</em>(<em>t</em>)</span> is reduced to its product expression. Perhaps this is close to the original idea of Riemann. Other results are also discussed by geometric analysis in the last section.展开更多
Using translation β = σ −1/2 and rotation s =σ + it = 1/2 + iz, z = t −iβ, Riemann got two results: (Theorem A) the functional equation ξ(z) = G(s)ξ(s), where , and (Theorem B) the product ex...Using translation β = σ −1/2 and rotation s =σ + it = 1/2 + iz, z = t −iβ, Riemann got two results: (Theorem A) the functional equation ξ(z) = G(s)ξ(s), where , and (Theorem B) the product expression , where z<sub>j</sub> are all roots of ξ(z), including complex roots. He proposed Riemann conjecture (RC): All roots of ξ(z) are real. As the product expression can only be used as a tool of contradiction, we prove RC by contradiction. To avoid the zeros of ξ(1/2 + it), define a subset . We have basic estimate , on L (R). One can construct by all real roots t<sub>j</sub> of ξ(t). If ξ has no complex roots, then w(t) = G(s)ξ(s) for s = 1/2 + it. If the product expression has a complex root z'=t' −iα, where 0 a ≤ 1/2, R' = |z′| > 10, then ξ(z) has four complex roots ±(t′ ± iα), and should contain fourth order factor p(z), i.e. ξ(z) = w(z)p(z). But p(z) can not be contained in ξ(s), as we have on L(R) and p(t) ≥ 0.5(t/R)<sup>4</sup> . As a result, we can rewrite ξ(t) = w(t)p(t) =G(s)ξ(s)p(t) on and get This contradicts the basic estimate. Therefore ξ(z) has no complex roots and RC holds.展开更多
Quantitative analysis of ras oncogene product P21 was performed on paraffin blocks from 55 smooth musele tumors of the gastrointestinal tract by immunofluorescence and flow cytometry.No positive evidence for P21 was f...Quantitative analysis of ras oncogene product P21 was performed on paraffin blocks from 55 smooth musele tumors of the gastrointestinal tract by immunofluorescence and flow cytometry.No positive evidence for P21 was found in 5 cases of normal smooth muscle tissues.展开更多
Objective To construct a eukaryotic expression system with pcDNA3-PfCSP/Hela for the Circumsporozoite protein (CSP) gene of Plasmodium falciparum (P. falciparum), to observe the immune responses in BALB/c mice induce...Objective To construct a eukaryotic expression system with pcDNA3-PfCSP/Hela for the Circumsporozoite protein (CSP) gene of Plasmodium falciparum (P. falciparum), to observe the immune responses in BALB/c mice induced by the expressed proteins. Methods The recombinant plasmid pcDNA3-PfCSP was transformed into the Hela cell line. The expressed protein was isolated and analyzed by using SDS-PAGE and used for immunization of BALB/c mice by subcutaneous, intravenous, and intraperitoneal adminstration. Enzyme-linked immunosorbent assay(ELISA), Dot-ELISA, Western blot, T lymphocyte proliferation test, natural killer cell(NKC) activity assay, and CD4+ and CD8+ T cell detection were used for observation of humoral and cellular immune responses. Results Immune sera strongly reacted with the expressed protein, antibody titer was up to 1∶6400 as detected by ELISA. Western blot analysis revealed a specific band at 38.3?Kda. When the spleen cells of normal and immunized BALB/c mice were specifically stimulated with expressed protein, the optical densities were 0.12±0.03 and 0.34±0.04, respectively. The latter were significantly higher than the former (P<0.01). We used the MTT colorimetric assay to measure NKC activity of mice spleen. The results showed that the NKC activity of immunized BALB/c mice was remarkably higher than that of the controls (P<0.05). CD4+ and CD8+ T cells were detected by using monoclonal antibody immunofluorescence methods. The results showed that the percentage of CD4+ and CD8+ T cells of immunized group were significantly higher than that of control group (P<0.05).Conclusions The humoral and cell-mediated immune responses and elevated NKC activity to products made with a eukaryotic expression system could be specifically detected in BALB/c mice. These findings indicate that the expressed protein could enhance the immune function in mice.展开更多
文摘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(s<sub>0</sub>)ζ (s<sub>0</sub>), s<sub>0</sub> =1/2 + it and (Theorem B) product expression ξ<sub>1</sub>(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 s<sub>0</sub> =1/2 + it. Using all real roots r<sub>k</sub><sub> </sub>and (true) complex roots z<sub>j</sub> = t<sub>j</sub> + ia<sub>j</sub> of ξ (z), define product expressions w(t), w(0) =ξ(0) and Q(t) > 0, Q(0) =1 respectively, so ξ<sub>1</sub>(t) = w(t)Q(t). Define infinite point-set L(ω) = {t : t ≥10 and |ζ(s<sub>0</sub>)| =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 |z<sub>1</sub>|>>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.
文摘IL-2 production and IL-2 receptor (Tac antigen) of the peripheral blood mononuclear cells in 30 patients with aplastic anemic (AA) were studied. We found that mononuclear cells from patients produce spontaneously IL-2 in the absence of exogenous lee-tin stimulation, the proportion of Tac+ cells in mononuclear cells increased. The release of IL-2 and or Tac antigen expression were elevated in almost every patient with AA. The plasma from patients stimulate mitogen-induced blastogenesis and Tac antigen expression of normal human lymphocytes. Immunological 1 abnormalities of patients with AA possibly might represents secondary response to bone marrow depression.
文摘Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that A<sup>n</sup> + B<sup>n</sup> = C<sup>n</sup>, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A<sup>2</sup> + B<sup>2</sup> + C<sup>2</sup> + D<sup>2</sup> + so on =A<sub>n</sub><sup>2 </sup>where all are natural numbers.
文摘Riemann (1859) had proved four theorems: analytic continuation ζ(s), functional equation ξ(z)=G(s)ζ(s)(s=1/2+iz, z=t−i(σ−1/2)), product expression ξ1(z)and Riemann-Siegel formula Z(z), and proposed Riemann conjecture (RC): All roots of ξ(z)are real. We have calculated ξand ζ, and found that ξ(z)is alternative oscillation, which intuitively implies RC, and the property of ζ(s)is not good. Therefore Riemann’s direction is correct, but he used the same notation ξ(t)=ξ1(t)to confuse two concepts. So the product expression only can be used in contraction. We find that if ξhas complex roots, then its structure is destroyed, so RC holds. In our proof, using Riemann’s four theorems is sufficient, needn’t cite other results. Hilbert (1900) proposed Riemann hypothesis (RH): The non-trivial roots of ζhave real part 1/2. Of course, RH also holds, but can not be proved directly by ζ(s).
文摘In this paper, we consider a general quasi-differential expressions t1,t2 Tn, each of order n with complex coefficients and their formal adjoints are t1＋,t2＋- x＋ on [0, b） respectively. We show in the direct sum spaces LZ（Ip）, p = 1,2 N of functions defined on each of the separate intervals with the case of one singular end-points and under suitable conditions on the function F that all solutions of the product quasi-integro differential equations are bounded and LZw -bounded on [0,b）.
文摘This paper will prove Riemann conjecture(RC): All zeros of <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em>ξ</em></span>(<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em>τ</em></span></span></span></span>)</span> lie on critical line. Denote <img src="Edit_189dc2b2-73ef-4036-9f06-ecf8a47fe58b.png" width="140" height="16" alt="" />, and <img src="Edit_a8ec55cb-e4c4-4156-ba23-ae01a31d1bc8.png" width="110" height="22" alt="" /> on critical line. We have found two mysteries in Riemann’s paper. <em>The first mystery</em> is the equivalence: <img src="Edit_3c075830-3c6c-4a23-9851-5b7d219e8000.png" width="140" height="21" alt="" /> is uniquely determined by its initial value <span style="white-space:nowrap;"><em>u</em> (<em>t</em>)</span>. <em>The second mystery</em> is Riemamm conjecture 2 (RC2): Using all zeros <span style="white-space:nowrap;"><em>t<sub>j</sub> </em></span>of <em>u</em> (<em>t</em>) can uniquely express <img src="Edit_b15d9c18-b55b-49e3-97a1-d2e03ccb6343.png" width="175" height="23" alt="" />. We find that the proof of RC is hidden in it. Our basic idea as follows. Consider functional equation <img src="Edit_f5295ff4-90b2-4465-851a-cad140b181c8.png" width="305" height="20" alt="" />. It is known that on critical line <img src="Edit_b45bff49-6d09-456b-9d1f-4259c66293d3.png" width="310" height="23" alt="" /> and <img src="Edit_4182ba79-0fcb-4f84-b7e7-c7574406596e.png" width="85" height="26" alt="" />, then we have the upper bound of growth <img src="Edit_d3d84d75-cc56-47b8-a9a7-ef8a9a5f07b1.png" width="250" height="33" alt="" /> To prove RC2 (or RC), by contradiction. If <span style="white-space:nowrap;"><em>ξ</em>(<em>τ</em>)</span> has conjugate complex roots <em>t</em>'<span style="white-space:nowrap;">±<em>i</em><span style="white-space:nowrap;"><em>β</em></span>'’</span>, <span style="white-space:nowrap;"><em>β</em>'>0</span>, <em>R</em><sup>2</sup>=t'<sup>2</sup>+<span style="white-space:nowrap;"><em>β</em>'<sup>2</sup></span>, by symmetry <span style="white-space:nowrap;"><em>ξ</em>(<em>τ</em>)=<span style="white-space:nowrap;"><em>ξ</em>(-<em>τ</em>)</span></span>, then -(<em>t</em>'<span style="white-space:nowrap;">±<em>i</em><em>β</em>''</span>) do yet. So <em>ξ</em> must contain four factors. Then <em>u</em>(<em>t</em>) contains a real factor <img src="Edit_ac03c1a5-0480-4efa-aac4-7788852a42a9.png" width="225" height="22" alt="" /> and <span style="white-space:nowrap;">ln|<em>u</em>(<em>t</em>)|</span> contains a term (the lower bound) <img src="Edit_6e94ad71-a310-4717-99ee-90384b0d89ba.png" width="230" height="19" alt="" /> which contradicts to the growth above. So <span style="white-space:nowrap;"><em>ξ</em></span> can not have the complex roots and <em>u</em>(<em>t</em>) does not have the factor <em>p</em>(<em>t</em>). Therefore both RC2 and RC are proved. We have seen that the two-dimensional problem is reduced to one-dimension and the one-dimensional <span style="white-space:nowrap;"><em>u</em>(<em>t</em>)</span> is reduced to its product expression. Perhaps this is close to the original idea of Riemann. Other results are also discussed by geometric analysis in the last section.
文摘Using translation β = σ −1/2 and rotation s =σ + it = 1/2 + iz, z = t −iβ, Riemann got two results: (Theorem A) the functional equation ξ(z) = G(s)ξ(s), where , and (Theorem B) the product expression , where z<sub>j</sub> are all roots of ξ(z), including complex roots. He proposed Riemann conjecture (RC): All roots of ξ(z) are real. As the product expression can only be used as a tool of contradiction, we prove RC by contradiction. To avoid the zeros of ξ(1/2 + it), define a subset . We have basic estimate , on L (R). One can construct by all real roots t<sub>j</sub> of ξ(t). If ξ has no complex roots, then w(t) = G(s)ξ(s) for s = 1/2 + it. If the product expression has a complex root z'=t' −iα, where 0 a ≤ 1/2, R' = |z′| > 10, then ξ(z) has four complex roots ±(t′ ± iα), and should contain fourth order factor p(z), i.e. ξ(z) = w(z)p(z). But p(z) can not be contained in ξ(s), as we have on L(R) and p(t) ≥ 0.5(t/R)<sup>4</sup> . As a result, we can rewrite ξ(t) = w(t)p(t) =G(s)ξ(s)p(t) on and get This contradicts the basic estimate. Therefore ξ(z) has no complex roots and RC holds.
文摘Quantitative analysis of ras oncogene product P21 was performed on paraffin blocks from 55 smooth musele tumors of the gastrointestinal tract by immunofluorescence and flow cytometry.No positive evidence for P21 was found in 5 cases of normal smooth muscle tissues.
文摘Objective To construct a eukaryotic expression system with pcDNA3-PfCSP/Hela for the Circumsporozoite protein (CSP) gene of Plasmodium falciparum (P. falciparum), to observe the immune responses in BALB/c mice induced by the expressed proteins. Methods The recombinant plasmid pcDNA3-PfCSP was transformed into the Hela cell line. The expressed protein was isolated and analyzed by using SDS-PAGE and used for immunization of BALB/c mice by subcutaneous, intravenous, and intraperitoneal adminstration. Enzyme-linked immunosorbent assay(ELISA), Dot-ELISA, Western blot, T lymphocyte proliferation test, natural killer cell(NKC) activity assay, and CD4+ and CD8+ T cell detection were used for observation of humoral and cellular immune responses. Results Immune sera strongly reacted with the expressed protein, antibody titer was up to 1∶6400 as detected by ELISA. Western blot analysis revealed a specific band at 38.3?Kda. When the spleen cells of normal and immunized BALB/c mice were specifically stimulated with expressed protein, the optical densities were 0.12±0.03 and 0.34±0.04, respectively. The latter were significantly higher than the former (P<0.01). We used the MTT colorimetric assay to measure NKC activity of mice spleen. The results showed that the NKC activity of immunized BALB/c mice was remarkably higher than that of the controls (P<0.05). CD4+ and CD8+ T cells were detected by using monoclonal antibody immunofluorescence methods. The results showed that the percentage of CD4+ and CD8+ T cells of immunized group were significantly higher than that of control group (P<0.05).Conclusions The humoral and cell-mediated immune responses and elevated NKC activity to products made with a eukaryotic expression system could be specifically detected in BALB/c mice. These findings indicate that the expressed protein could enhance the immune function in mice.
