Orientifolds and Slumps in G_2 and Spin(7) Metrics

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SYSTEM AND METHOD FOR ISOLATING AND ANALYZING CELL

专利名称：SYSTEM AND METHOD FOR ISOLATING AND ANALYZING CELLS发明人：HANDIQUE, Kalyan,SHARMA, Vishal,GOGOI, Priyadarshini,CHOW, William,PAYNE,Austin,BONIFACE, Brian,GLEASON,Kyle,CONNOLLY, John,TUCK, Sam申请号：US2018/048353申请日：20180828公开号：WO2019/046307A1公开日：20190307专利内容由知识产权出版社提供专利附图：摘要：A system and method for isolating and analyzing single cells, wherein the system includes: an array of wells defined at a substrate, each well including an open surface and a well cavity configured to capture cells in one of a single-cell format and single-cluster format, and a fluid delivery module including a fluid reservoir superior to the array of wells through which fluid flow is controlled along a fluid path in a direction parallel to the broad face of the substrate; and wherein the method includes: distributing a population of cells and a population of non-cell particles across the array of wells through the fluid reservoir to increase capture efficiency of individual cell-particle pairs within the array of wells, and processing the captured cell-particle pairs at the set of wells.申请人：CELSEE DIAGNOSTICS, INC.地址：46701 Commerce Center Drive Plymouth, MI 48170 US国籍：US代理人：SCHOX, Jeffrey更多信息请下载全文后查看。

Method and device for simultaneously determining t

专利名称：Method and device for simultaneouslydetermining the adhesion, friction, and othermaterial properties of a sample surface发明人：Hans-Ulrich Krotil,Thomas Stifter,OthmarMarti申请号：US09869789申请日：20000104公开号：US06880386B1公开日：20050419专利内容由知识产权出版社提供专利附图：摘要：A process for the location-resolved simultaneous detection of the adhesion andfriction as well as possibly of other material properties of a sample surface to be examined by means of a raster probe microscope comprising a raster probe. The raster probe and/or the sample with sample surface are moved until at a point of the sample surface to be examined the raster probe interacts in a determined manner with this surface. The raster probe and/or the sample are subjected to a vertical oscillation, and a first measuring signal characterized by the deformation of the raster probe is recorded.A second measuring signal characterizing the deformation of the raster probe is recorded, wherein the raster probe and/or the sample are subjected to a horizontal and/or vertical oscillation. From these two measuring signals the desired material properties are determined. For the detection of the entire surface area to be examined the raster probe and or the sample are again moved and for the repetition of the measuring process described brought into contact with the sample surface in the above described manner.申请人：Hans-Ulrich Krotil,Thomas Stifter,Othmar Marti地址：Neu-Ulm DE,Illereichen DE,Ulm DE国籍：DE,DE,DE代理机构：Baker & Daniels更多信息请下载全文后查看。

Biosensors and Bioelectronics

Biosensors and Bioelectronics Biosensors and bioelectronics have revolutionized the field of medical diagnostics and have played a crucial role in the advancement of personalized medicine. These technologies have enabled the development of highly sensitive and specific devices for the detection of biological molecules, such as proteins, nucleic acids, and small molecules, in various biological samples. The ability to accurately measure and monitor biomarkers in real-time has significantimplications for disease diagnosis, monitoring, and treatment. In this response, we will delve into the historical background, perspectives, case studies, critical evaluation, and future implications of biosensors and bioelectronics. The development of biosensors and bioelectronics can be traced back to the early 1960s when Leland C. Clark and Champ Lyons first introduced the concept of the enzyme electrode for the measurement of oxygen. This pioneering work laid the foundation for the development of enzyme-based biosensors, which are now widely used in clinical diagnostics, environmental monitoring, and food safety. Over the years, significant advancements have been made in the design and fabrication of biosensors, leading to the development of various types of biosensors, including electrochemical, optical, and piezoelectric biosensors. From a historical perspective, the field of biosensors and bioelectronics has witnessed rapid growth and evolution, driven by advances in nanotechnology, microfabrication, and biotechnology. The integration of nanomaterials, such as carbon nanotubes, graphene, and metal nanoparticles, has significantly enhanced the sensitivity and selectivity of biosensors. Furthermore, the miniaturization of biosensor devices has enabled point-of-care testing and wearable biosensors, offering unprecedented opportunities for real-time health monitoring and personalized medicine. From an analytical perspective, biosensors and bioelectronics have garnered widespread attention and interest from researchers, clinicians, and industry stakeholders. The ability to detect and quantify biomarkers with high sensitivity andspecificity has immense implications for disease diagnosis and management. For example, biosensors have been developed for the early detection of cancer biomarkers, monitoring of glucose levels in diabetes patients, and detection of infectious diseases such as COVID-19. These advancements have the potential torevolutionize healthcare by enabling early intervention and personalized treatment strategies. In addition to healthcare applications, biosensors and bioelectronics have also found diverse applications in environmental monitoring, food safety, and bioprocess monitoring. For instance, biosensors have been developed for the detection of environmental pollutants, such as heavy metals and pesticides, enabling rapid and on-site analysis. In the food industry, biosensors are used for the detection of foodborne pathogens and contaminants, ensuring the safety and quality of food products. Moreover, biosensors have been integrated into bioreactors for real-time monitoring of fermentation processes, optimizing bioproduction and bioprocessing. Despite the significant advancements and potential applications of biosensors and bioelectronics, there are certain challenges and limitations that need to be addressed. One of the key challenges is the need for standardization and validation of biosensor technologies to ensure their reliability and reproducibility. Additionally, the integration of biosensors into clinical practice and healthcare systems requires overcoming regulatory and economic barriers. Furthermore, the translation of biosensor technologies from the laboratory to the market necessitates interdisciplinary collaboration and investment in research and development. In conclusion, biosensors and bioelectronics have emerged as powerful tools for the detection and monitoring of biological molecules, with far-reaching implications for healthcare, environmental monitoring, and biotechnology. The integration of advanced nanomaterials, microfabrication techniques, and biorecognition elements has enabled the development of highly sensitive and specific biosensors for diverse applications. However, addressing the challenges related to standardization, validation, and translation of biosensor technologies is crucial for their widespread adoption and impact. Looking ahead, continued research and innovation in biosensors and bioelectronics will pave the way for novel diagnostic and monitoring technologies, ultimately contributing to improved healthcare and sustainability.。

细菌双组分调节系统 Two-Component Regulatory System

四种常用抗生素对大肠杆菌sos反应的诱导意义?是细菌的一种普遍而又重要的信号传递系统也是一个基因表达的调节系统?理论意义就在于阐明了细菌对环境变化做出反应的基本过程为深化细菌行为生态生理生化的基本过程为深化细菌行为生态生理生化研究奠定了基础?二元信号传递的作用对象是那些赋予细菌具有特殊功能行为的基因
TCSTS）
Two-component regulatory systesms
1. 基本所有细菌都有: 1% of genome 2. 有信号识别和传导（激酶）两部分 3. RR通常为转录调节蛋白
HPK是一种跨膜蛋白，几乎所有的HPK都含有2个跨膜区（TM1、TM2）。HPK的N-端有一个能感受外界信号的输入区，C-端有一个由约250个氨基酸残基组成的转导区，该区具有自主磷酸激酶的功能，磷酸化的位点一般是保守的His残基（H）。此外，转导区中还含有5个由5～10个氨基酸残基组成的高度保守区域。
本节内容总结
腺苷脱腺苷控制谷氨酰胺合成酶（GS）活性。氮信号被腺（尿）苷酰转移酶（ATase或 UTase）识别。低氮激活UTase，尿苷酰化的PII协助ATase对GS-AMP 去腺苷酰化。
细菌群体感应
抗生素诱细菌SOS反应
Kohanski MA, 201
细菌的SOS 反应
ቤተ መጻሕፍቲ ባይዱ
四种常用抗生素对大肠杆菌SOS反应的诱导
SOS反应诱导错误倾向的DNA复制，大大提高碱基自发突变率，导致产生多重抗生素耐药性； SOS反应通过诱导整合子重组，促进溶源性噬菌体的裂解，加速了耐药基因在细菌间的水平转移[18]。另外，SOS反应诱导蛋白可以提高细菌对抗生素的耐药性，比如链球菌（Streptococcus pneumoniae）热休克蛋白 ClpL的诱导表达可以增强链球菌对青霉素的耐药性。

The complex separation and extensions of Rokhlin congruence for curves on surfaces

a r X i v :a l g -g e o m /9206009v 1 24 J u n 1992The complex separation and extensions of Rokhlin congruence for curves on surfacesGrigory MikhalkinAbstract The subject of this paper is the problem of arrangement of real algebraic curves on real algebraic surfaces.In this paper we extend Rokhlin,Kharlamov-Gudkov-Krakhnov and Kharlamov-Marin congruences for curves on surfaces and give some applications of this extension.For some pairs consisting of a surface and a curve on this surface (in particular for M-pairs)we introduce a new structure —the complex separation that is sepa-ration of the complement of curve into two surfaces.In accordance with Rokhlin terminology the complex separation is a complex topological characteristic of real algebraic varieties.The complex separation is similar to complex orientations in-troduced by O.Ya.Viro (to the absolute complex orientation in the case when a curve is empty and to the relative complex orientation otherwise).In some cases we calculate the complex separation of a surface (for example in the case when surface is the double branched covering of another surface along a curve).With the help of these calculations applications of the extension of Rokhlin congruence gives some new restrictions for complex orientations of curves on a hyperboloid.1Introduction 1.1Rokhlin and Kharlamov-Gudkov-Krakhnov congruences for curves on surfaces If A is a real curve of odd degree on real projective surface B then A divides B into twoparts B +where polynomial determining A is non-negative and B −where this polynomial is non-positive.V.A.Rokhlin [1]proved the congruence for the Euler characteristic of B −under such strong hypotheses that they follow that B is an M-surface,A is an M-curve and B +is contained in a connected component of B .V.M.Kharlamov [2],D.A.Gudkov and A.D.Krakhnov [3]proved a relevant congruence that makes sense sometimes even if A is not M-but (M-1)-curve,but the hypotheses for B and B +were not weakened.1.2Description of the paperThe results of the paper make sense in the case when a pair consisting of a surface and a curve in this surface is of characteristic type (for deﬁnition see section 2).For pairs of characteristic type we introduce a complex separation of the complement of the curve in the surface that is a new complex topological characteristic of pairs consisting of asurface and a curve in this surface.If the curve is empty then the complex separation is a new complex topological characteristic of surfaces relevant to the complex orientation of surfaces introduced by O.Viro[4].The main theorem is formulated with the help of the complex separation,this theorem is a generalization of Rokhlin and Kharlamov-Gudkov-Krakhnov congruences for curves in surfaces.The main theorem gives nontrivial restrictions even for curves of odd degree on some surfaces.The paper contains also some applications of the main theorem.We prove a congruence modulo32for Euler characteristic of real connected surface of characteristic type.We prove some new congruences for curves on a hyperboloid.We give a direct extension of Rokhlin and Kharlamov-Gudkov-Krakhnov congruences for curves on projective surfaces, there we avoid both of the hypotheses that B is an M-surface and B+is contained in a connected component of B.We apply the main theorem to the classiﬁcation of curves of low degrees on an ellipsoid.In particular,we get a complete classiﬁcation ofﬂexible curves on an ellipsoid of bidegree(3,3)(the notion ofﬂexible curve is analogous to one introduced by O.Viro[5]for plane case).One can see that the fact that the theorem can be applied to curves of odd degree follows that this theorem can not be proved in the Rokhlin approach using the double covering of the complexiﬁcation of the surface branched along the curve(since there is no such a covering for curves of odd degree).We use the Marin approach[6].All results of this paper apply toﬂexible curves as well as to algebraic.The author is indebted to O.Ya.Viro for his attention to the paper and consultations. 2Notations and the statement of the main theorem Let C B be a smooth oriented4-manifold such that itsﬁrst Z2-Betti number is zero equipped with an involution conj such that the set R B of itsﬁxed points is a surface. Let C A be a smooth surface in C B invariant under conj and such that the intersection of C A and R B is a curve.These notations are inspired by algebraic geometry.It is said that A is of even degree if C A is Z2-homologous to zero in C B and that A is of odd degree otherwise.It is said that curve A is of type I if R A is Z2-homologous to zero in C A and that A is of type II otherwise.It is said that surface B is of type I abs if R B is Z2-homologous to zero in C B If B is a real projective plane then it is said that B is of type I rel if R B is Z2-homologous to a plane section of C B.We shall say that pair (B,a)is of characteristic type if the sum of R B and C A is Poincar´e dual to the second Stiefel-Whitney class of C B.We shall say that surface B is of characteristic type if R B is a characteristic surface in C B.Let b∗denote the total Z2-Betti number.It is said that manifold C X equipped with involution conj is an(M-j)-manifold if b∗(R X)+2j=b∗(C X)where R X is theﬁxed point set of conj.One can easily see that Smith theory follows that j is a nonnegative integer number.Letσ(M)denote the signature of oriented manifold M.;D M:H∗(M;Z2)→H∗(M;Z2)denote Poincar´e duality operator;[N]∈H∗(M;Z2)denote Z2-homology class of submanifold N⊂M.Let e A=[C A◦C A]C B denote normal Euler number of R B in C B.If Bǫis a surface contained in R B and such that∂Bǫ=R A that we shall denote by e Bthe obstruction to extending of line bundle over R A and normal to R A in C A to ǫthe line bundle over Bǫnormal in C B to R B evaluated on the twisted fundamental class[Bǫ,∂Bǫ]and divided by2.One can see that if(B,A)is a nonsingular pair consisting of an algebraic surface and algebraic curve then e Bǫ=−χ(Bǫ).Letβ(q)denote the Brown invariant of Z4-valued quadratic form q.Theorem1If(B,A)is of characteristic type then there is a natural separation of R B−R A into surfaces B1and B2such that∂B1=∂B2=R A deﬁned by the condition that C A/conj∪B j is a characteristic surface in C B/conj(j=1,2).There is a congruence for the Guillou-Marin form q j on H1(C A/conj∪B j;Z2)e Bj ≡e R B+σ(C B)4−β(q j)(mod8)Addendum1Let q j|H1(R A;Z2)=0a)If A is an M-curve then e Bj ≡e R B+σ(C B)4−βj(mod8)b)If A is an(M-1)-curve then e Bj ≡e R B+σ(C B)4−βj±1(mod8)c)If A is an(M-2)-curve and e Bj ≡e R B+σ(C B)4−βj+4(mod8)then A isof type Id)If A is of type I then e Bj ≡e R B+σ(C B)4(mod4)whereβj is the Brown invariant of the restriction q j|H1(B j;Z2)2.1RemarkSome of components of R A can be disorienting loops in C A(it is easy to see that number of such components is even).Ifαis some1-dimensional Z2-cycle in B j that is a boundary of some2-chainβin R B containing an even number of disorienting C A components of R A then q j(α)=0;if such a number is odd then q j(α)=2.It follows that if(B,A) is of characteristic type then the number of disorienting C A components of R A in each component of R B is even.3The proof of Theorem1and Addendum13.1Calculation of the characteristic class of C B/conjIt is not diﬃcult to see the formula for the characteristic classes of double branched covering:ifπ:Y→X is a double covering branched along Z thenw2(Y)=π∗w2(X)+D−1Y[Z]Applying this formula to p:C B→C B/conj we gettr(Dw2(C B/conj))=Dw2(C B)+[R B]where tr:H2(C B/conj;Z2)→H2(C B;Z2)is transfer(i.e.the inverse Hopf homomor-phism to P).It is easy to see that transfer can be decomposed as the composition H2(C B/conj;Z2)k→H2(C B/conj,R B;Z2)h→H2(C B;Z2)where k is an inclusion homomorphism and h◦k=tr.To prove that h is a monomorphism we use the Smith exact sequence(see e.g.[7]):H3(C B/conj,R B;Z2)γ3→H2(R B;Z2)⊕H2(C B/conj,R B;Z2)α2→H2(C B;Z2)In this sequence theﬁrst component ofγ3is equal to the boundary homomorphism ∂of pair(C B/conj,R B);∂is a monomorphism since H3(C B/conj;Z2)=0(since C B and therefore C B/conj are simply connected).It means that no element of type (0,x)∈H2(R B;Z2)⊕H2(C B/conj,R B;Z2),x=0is contained in Imγ3and there-fore the restriction ofα2to H2(C B/conj,R B;Z2)is a monomorphism and thus H is a monomorphism.Now if[C A]=Dw2(C B)+[R B]thenDw2(C B/conj)=[C A/conj]∈H2(C B/conj,R B;Z2)The exactness of homology sequence of pair(C B/conj,R B)follows since H is a monomor-phism that there exists a surface B1⊂R B such that∂B1=R A and W1=B1∪C A/conj is dual to w2(C B/conj).Let B2be equal to Cl(R B−B1)Surface R B is Z2-homologous to zero in C B/conj since C B is a double covering of C B/conj branched along R B.It follows that W2=B2⊕C A/conj is also a surface dual to w2(C B/conj).Note that the separation of R B into B1and B2is unique sincedim(ker(in∗:H2(R B;Z2)→H2(C B/conj)))=1as it follows from exactness of the Smith sequence.Indeed,since H3(C B/conj;Z2)=0 then this dimension is equal to the dimension of H3(C B/conj,R B;Z2);the dimen-sion of H3(C B/conj,R B;Z2)is equal to1sinceγ4:H4(C B/conj,R B;Z2)→0⊕H3(C B/conj,R B;Z2)is an isomorphism.3.2Proof of the congruence forχ(B j)Note that since W j,j∈{1,2}is a characteristic surface in C B/conj and C B is simply connected we can apply the Guillou-Marin congruence to pair(C B/conj,W j)σ(C B/conj)≡W j◦W j+2β(q j)(mod16)where q j:H1(W j;Z2)→Z4is the quadratic form associated to the embedding of W j into C B/conj(see[8]).Similar to the calculations in[6]we get thate AW j◦W j=2Combining all this we getχ(B j)≡e A4+β(q j)(mod8)Now if q|H1(R A;Z2)=0then additivity of the Brown invariant(see[9])follows thatβ(q j)=β(q|H1(C A/conj;Z2))+βjIt is easy to see that if A is an(M-j)-curve then rkH1(C A/conj;Z2)=j.Points a)and b)of the addendum immediately follow from this.To deduce points c)and d)of theaddendum note that q|H1(C A/conj;Z2)is even iﬀC A/conj is an orientable surface iﬀA isof type I(cf.[6],[9]).4Some applications of Theorem14.1The case C A is empty;congruences for surfaces4.1.1If Dw2(C B)=[R B]then there is deﬁned a complex separation of R B into two closed surfaces B1and B2;there is deﬁned a Z4-quadratic form q on H1(R B;Z2)=H1(B1;Z2)⊕H1(B2;Z2)equal to sum of Guillou-Marin forms of B1and B2which are characteristic surfaces in C B/conj andχ(B j)≡χ(R B)−σ(C B)It follows from an easy observation thatχ(B j)≡β(q|H1(B j;Z2))(mod2)4.1.4RemarkAccording to O.Viro[5]in some cases one can deﬁne some more complex topological char-acteristics on R ly,If B is of type I abs then R B possesses two special reciprocal orientations(so-called semi-orientation)and a special spin structure.If Dw2(C B)=[R B] then R B possesses the special P in−-structure corresponding to Guillou-Marin form q C B: H1(R B;Z2)→Z4of surface R B in C B.The complex separation is a new topological characteristic for surfaces and4.1.1may be interpreted as a formula for this characteristic. Quadratic form q is not a new complex topological characteristic.4.1.5Form q is equal to q C B in the case when these forms are deﬁned(i.e.when Dw2(C B)= [R B])To prove this one can note that the index of a generic membrane in C B bounded by curves in R B diﬀers from the index of the image of this membrane in C B/conj by number of intersection points of this membrane and R B.4.1.6RemarkIf B is a complete intersection in the projective space of hypersurfaces of degrees m j,j= 1,...,s then the condition that Dw2(C B)=[R B]is equivalent to the condition that B is of type I abs in the case when s j=1m j≡0(mod2)and to the condition that B is of type I rel in the case when s j=1m j≡1(mod2).4.2Calculations for double coveringsWe see that to apply Theorem1one needs to be able to calculate the complex separation and the corresponding Guillou-Marin form.We calculate them in some cases in this subsection.By the semiorientation of manifold M we mean a pair of reciprocal orientations of M (note that this notion is nontrivial only for non-connected manifolds).It is easy to see that two semiorientations determine a separation of M;this separation is a diﬀerence of two semiorientations,namely,two components of M are of the same class of separation iﬀthe restrictions of the semiorientations on these components are the same.Let C B be the double covering of surface C X branched along curve C D invariant under the complex conjugation conj X in C X.Suppose C D is of type I.Let C D+be one of two components of C D−R D.Then R D possesses a special semiorientation called the complex semiorientation(see[10]).The invariance of C D under conj X follows that conj X can be lifted in two diﬀerent ways into an involution of C B.Let conj B be one of these two lifts.Let X−=p(R B)),X+=R X−int(X−)where p is the covering map.4.2.1Suppose that Dw2(C B)=[R B].Then for every component C of X+the complex sep-aration of∂C induced from complex separation of R B via p(namely,two circles of∂C are of the same class of separation iﬀthe lie in the image under p of the same class of the separation of R B)is equal to the diﬀerence of the semiorientations on∂C induced by the complex semiorientation of R D and the unique(since C is connected)semiorientation of C.In particular C is orientable.Proofequal to F′∪conj X F′.Then p−1(F)gives an element of H2(C B)(the construction of this element was suggested by O.Viro[11]).Since p−1(F)is invariant under conj B it gives an element in H2(C B/conj B;Z2),say f∈H2(C B/conj B;Z2).Let us calculate the self-intersection number of f.It is easy to see that because of symmetry the self-intersection number of f in C B/conj B is equal to the self-intersection number ofγδin C∪C D+. The deﬁnition of complex semiorientation follows that the self-intersection number ofγδin C∪C D+(and therefore the self-intersection number of f)is equal to zero in theﬁrst case and to one in the second case.4.2.2RemarkIn the case when R X is connected4.2.1completely determines the complex separation of R B.4.2.3(O.Viro[11])Ifλis a loop in R B such that p(λ)=∂(G),where G⊂R X thenq C B(λ)≡2χ(G∩X+)(mod4)4.2.4(O.Viro[11])Supposeγis a path in X−connecting points Qαand Qβof componentsαandβof R D respectively.Then q C B(p−1(γ))=0if the intersection numbers ofγwithαandβare of opposite sign(case1ofﬁg.1)and q C B(p−1(γ))=2otherwise(case2ofﬁg.1)4.3New congruences for complex orientations of curves on ahyperboloidIn this subsection we apply the results of4.1and4.2to double branched coverings over simplest surfaces of characteristic type,a plane and a hyperboloid.To state congruences it is convenient to use the language of integral calculus based on Euler characteristic developed by O.Viro[12].Let R A be a curve of type I in the connected surface R X.We equip R A with one of two complex orientations andﬁx X∞–one of the components of R X−R A.If R X−X∞is orientable and R A is R-homologous to zero for some ring R of coeﬃcient coeﬃcient then there is deﬁned function ind R:R X−R A→R equal to zero on X∞and equal to the R-linking number with oriented curve R A in R X−X∞otherwise.It is easy to see that ind R is measurable and deﬁned almost everywhere on R X with respect to Euler characteristic.Evidently,the function ind R2:R X−R A→R⊗R does not depend on the ambiguity in the choice of one of two complex orientation of R A.4.3.1Consider the case when X=P2,R=Z.Let A be a plane nonsingular real curve of type I given by polynomial f A of degree m=2k.Then R A is Z2-homologous to zero.Let X∞be the only nonorientable component of R P2−R A.Without loss of generality suppose that f A|X∞<0.Deﬁne X±to be equal to{y∈R P2|±f A(y)≥0}.Let p:C B→C P2be the double covering of C P2branched along C A(note that such a covering exists and is unique since m is even and C P2is simply connected).Let conj B:C B→C B be the lift of conj:C P2→C P2such that p(R B)=X−(as it is usual we denote F ix(conj B by R B).Lemmae6.6and6.7of[7]immediately follow that Dw2(C B)=R B.Therefore we can apply4.1.1and4.2.1.Proposition4.2.1follows that p−1(cl(ind Z2−1(2+4Z)))is equal to one of two surfaces of the complex separation of R B.Set B1to be equal to p−1(cl(ind Z−1Z2(2+4Z))),note that B is orientable since ind Z|X∞=0.4.3.2Lemmaβ(q|H1(B1;Z2))≡4χ(ind Z−1(3+8Z)∪ind Z−1(−3+8Z))(mod8)Proof4+β(q|H1(B1;Z2))(mod8)Note that p(R B)=ind Z−1(2Z),χ(ind Z−1(2Z))=1−χ(1+2Z)andσ(C B)=2−2k2. We get4χ(ind Z−1(2+4Z))≡1−χ(ind Z−1(1+2Z))−1+k2+8χ(ind Z−1(±3+8Z))(mod16) We can reformulate this in integral calculus languageR P2ind Z2dχ≡k2(mod16)Thus for projective plane we get nothing new but the reduction modulo16of the Rokhlin congruence for complex orientation[10]R P2ind Z2dχ=k24.3.4Consider now the case when X=P1×P1.Let A be a nonsingular real curve of type I in P1×P1of bidegree(d,r),i.e.the bihomogeneous polynomial f A determining A is of bidegree(d,r)where d and r are even numbers.Let X∞be a component of R P2−R A,X±={y∈R P1×R P1|±f A(y)≥0}.Suppose without loss of generality that f A|X∞<0.Let p:C B→C P2be the double covering of C P2branched along C Aand let conj B:C B→C B be the lift of conj:C P1×C P1→C P1×C P1such that p(R B)=X−.Nonsingularity of A follows that all components of R A non-homologous to zero are homologous to each other.Let e1,e2form the standard basis of H1(R P1×R P1) and let s,t be the coordinates in this basis of a non-homologous to zero component of R A equipped with such an orientation that s,t≥0.If all the components of R A are homologous to zero then set s=t=0.Then R A equipped with the complex orientation produces l′(se1+te2)in H1(R P1×R P1).Note that s and t are relatively prime and l′is even since both d and r are even.4.3.5LemmaIf l′≡0(mod4)then Dw2(C B)=[R B]The proof follows from Lemma3.1of[13].4.3.6LemmaIf l′≡0(mod4)then ind Z4is deﬁned and if sd+tr≡0(mod4)thenR P1×R P1ind Z42dχ≡drNote that the condition that sd+tr≡0(mod4)is just equivalent to the ori-entability of R B.Thereforeβ(q|H1(B1;Z2))≡0(mod4).Further arguments are similarto the plane case,we skip them.4.3.7RemarkThe traditional way of proving of formulae of complex orientations for curve on surfaces (see[15])adjusted to the case when the real curve is only Z4-homologous to zero givesonly congruenceR P1×R P1ind Z42dχ≡dr2(mod8) 4.3.8If l′≡4(mod8)and sd+tr≡2(mod4)thenR P1×R P1ind Z42dχ≡drForm q|H1(B1;Z2)is cobordant to the sum of a form on an orientable surface andsome forms on Klein bottles.The condition that l′≡4(mod8)is equivalent to the condition that the number of forms on Klein bottles non-cobordant to zero is odd.Thusβ(q|H1(B1;Z2))≡2(mod4)4.3.9If l′≡0(mod8)and sd+tr≡0(mod4)thenR P1×R P1ind2Z8dχ≡dr2t+r2(mod8)b)If A is an(M-1)-curve thenχ(B+)≡dr2+4(mod8)then A is of type Id)If A is of type I thenχ(B+)≡0(mod4)This theorem follows from Theorem1and gives some new restriction on the topology of the arrangement of real nonsingular algebraic curve of even bidegree on a hyperboloidwith non-contractible branches.Points a)and b)of4.3.10in the case when d2s≡0(mod2),s+t≡1(mod2)were proved by S.Matsuoka[14]in another way(using 2-sheeted branched coverings of hyperboloid).Point d)of4.3.10is a corollary of the modiﬁcation of Rokhlin formula of complex orientations for modulo4case.4.4The case when A is a curve of even degree on projectivesurface BIn this subsection we deduce Rokhlin and Kharlamov-Gudkov-Krakhnov congruences from Theorem1and give a direct generalization of these congruences.Let B be the surface in P q given by the system of equations P j(x0,...,x q)=0,j= 1,...,s−1,let A be the(M-k)-curve given by the system of equations P j(x0,...,x q)= 0,j=1,...,s−1,where P j are homogeneous polynomials with real coeﬃcients,degP j= m j,s=q−1.Suppose(B,A)is a non-singular pair,R A=∅and m s is even.Denote B+={x∈R B|P s(x)≥0},B−={x∈R B|P s(x)≤0},d=rk(in B+∗:H1(B+;Z2)→H1(R B;Z2)),e=rk(in R A∗:H1(R A;Z2)→H1(R B;Z2)) Set c to be equal to the number of non-contractible in R P q components of R B not intersecting R A.Let us reformulate Rokhlin and Kharlamov-Gudkov-Krakhnov congruences for curve on surfaces in the form convenient for generalization and correcting the error in[1].4.4.1(Rokhlin[1],Kharlamov[2],Gudkov-Krakhnov[3])Suppose B is an M-surface,e=0,B+is contained in one component of R B and in the case when m s≡0(mod4)suppose in addition that c=0.a)If d+k=0thenm1...m s−1m2sχ(B+)≡±1(mod8)4Indeed,it is easy to see that the hypothesis of4.4.1without the condition on c is equivalent to the hypotheses of corresponding theorems in[1],[2]and[3],to reformulate them it is enough to apply the Rokhlin congruence for M-surfaces.4.4.2Remark(on an error in[1])Point2.3of[1]contains a miscalculation of characteristic class x of the restriction to B−of the double covering of C B branched along C A.In[1]it is claimed that if m s≡2 (mod4)then x=w1(B−−A).It led to the omission of the condition on c in both Rokhlin and Kharlamov-Gudkov-Krakhnov congruences.4.4.3Correction of the error in[1]If m s≡2(mod4)then x=in∗αwhere in is the inclusion of B−into R P q andαis the only non-zero element of H1(R P q;Z2)Proofa)If d+k=0thenm1...m s−1m2sχ(B+)≡±1(mod8)4c)If d+k=2andm1...m s−1m2sχ(B+)≡(mod4)44.4.5LemmaThe surface V+=C A/conj∪B+is Z2-homologous to zero in C B/conjpProof-?where p:C Y→C B is the double covering over C B branched along C A and conj Y is such a lift of conj:C B→C B that R Y={y∈C Y|conj Y y=y}is equal to p−1(B−). It is easy to see that this diagram can be expanded to a commutative one by adding φ:C Y/conj Y→C B/conj whereφis the double covering map branched along V+.It follows that[V+]=0∈H2(C B/conj).4.4.6RemarkThe construction of4.4.5allows to deﬁne the separation of R B into B+and B−for any (not necessarily algebraic)curve of even degree and invariant with respect to conj in any (not necessarily projective)complex surface C B equipped with almost antiholomorphic involution conj:C B→C B and such that H1(C B;Z2)=0.In this case there is the unique double covering over C B branched along C A and two possible lifts of conj:C B→C B.The images of theﬁx point sets of these lifts form the desired separation.Note that this separation is diﬀerent from the complex separation.Let us give an internal deﬁnition of this separation.Let x,y be points in R B−R A.Connect these points by pathγinside C B−C A.If the linking number of the loopγconjγand C A is equal to zero then x and y are points of the same surface of the complex separation, otherwise x and y are points of two diﬀerent surfaces of the complex separation.This remark allows to extend4.4.4to the case ofﬂexible curves.4.4.7The proof of4.4.4Note that Dw2(C B)+[R B]=0since Dw2(C B)≡[∞] s−1j=1m j where[∞]∈H2(C B;Z2) is the class of hyperplane section of C B and note that[C A]=0since m s is even.We apply4.1.1,let W be the surface of the complex separation not intersecting B+. Then4.4.5follows that W∪B+is the surface of the complex separation of(B,A)since if W is a characteristic surface in C B/conj then so is W+=W∪V+.Let us prove now that q W+|H1(W;Z2)is equal to q W where q W+and q W are Guillou-Marin forms of W+and W in C B/conj.Note that(q W+−q W)(x),x∈H1(W;Z2)is equal to the linking number of x and V+in C B/conj that is equal to the linking number of x and C A in C B.It is not diﬃcult to see that the condition on c follows that the linking number of x and C A in C B is equal to zero.Theorem1follows thatχ(B+)+χ(W)≡e A4+β(q W)+β(q W+|H1(W+;Z2))(mod8)and4.1.1follows thatχ(W)≡χ(R B)−σ(C B)4+β(q W+|H1(V+;Z2))(mod8)The rest of the proof is similar to that of3.2.4.5Curves on an ellipsoidIn the congruences of4.4.4we was able to avoid the appearance of the complex separation with the help of the separation of R B into B+and B−.For curves of odd degree this does not work since there is no such a separation.Although,if R B is connected then the complex separation does not provide the additional information and still can be avoided. Surfaces B1and B2of the complex separation are determined by the condition that B1∪B2=R B and∂B1=∂B2=R A.The simplest case is the case of curves of odd degree on an ellipsoid.It is well-known that the complex quadric is isomorphic to C P1×C P1,an algebraic curve in quadric is determined by a bihomogeneous polynomial of bidegree(d,r).If the curve is real and the quadric is an ellipsoid then d=r,otherwise the curve can not be invariant under the complex conjugation of an ellipsoid since the complex conjugation of an ellipsoid acts on H2(C P1×C P1=Z×Z in the following way:conj∗(a,b)=(−b,−a) as it is easy to see considering the behavior of conj on the generating lines of C P1×C P1. Thus a real curve on an ellipsoid is the intersection of the ellipsoid and a surface of degree d,this can be regarded as a deﬁnition of real curves on an ellipsoid.4.5.1TheoremLet A be a nonsingular real curve of bidegree(d,d)on ellipsoid B.Suppose that d is odd.a)If A is an M-curve thenχ(B1)≡χ(B2)≡d2+12±1(mod8)χ(B2)≡d2+12+4(mod8) then A is of type Id)If A is of type I thenχ(B1)≡χ(B2)≡1(mod4) Proof4+χ(R B)−σ(C B)2andβj=0because of the triviality of H1(R B;Z2).4.5.2Low-degree curves on an ellipsoidConsider the application of4.5.1to the low-degree curves on an ellipsoid.Gudkov and Shustin[16]classiﬁed real schemes of curves of bidegree not greater then(4,4)on an ellipsoid.To prove restrictions for such a classiﬁcation it was enough to apply the Harnack inequality and Bezout ing the Bezout theorem avoided the extension of such a classiﬁcation to theﬂexible curves.For the curves of bidegree(4,4)one can avoid the using of the Bezout theorem using instead the analogue of the strengthened Arnold inequalities for curve on an ellipsoid.for the classiﬁcation of theﬂexible curves of bidegree(3,3)the old restrictions does not give the complete system of restrictions,they do not restrict schemes2⊔1<2>,3⊔1<1> and2⊔1<1>(see[5]for the notations).Theorem4.5.1restricts these schemes and thus completes the classiﬁcation of the real schemes ofﬂexible curves of bidegree(3,3)on an ellipsoid.4.5.3TheoremThe real scheme of a nonsingularﬂexible curve of bidegree(3,3)on an ellipsoid is1< 1<1>>orα⊔1<β>,α>β,α+β≤4.Each of this schemes is the real scheme of some ﬂexible(and even algebraic)curve of bidegree(3,3)on an ellipsoid4.5.4Curves of bidegree(5,5)on an ellipsoidTheorem4.5.1gives an essential restriction for the M-curves of bidegree(5,5)on an ellip-soid,4.5.1leaves unrestricted18possible schemes ofﬂexible M-curves of bidegree(5,5) and15of them can be realized as algebraic M-curves given by birational transforma-tions of the appropriate aﬃne curves of degree6and another one by Viro technique of small perturbation from the product ofﬁve plane sections intersecting in two diﬀerent points.Thus there are only two schemes of M-curves of bidegree(5,5)unrestricted and unconstructed,namely,1⊔1<6>⊔1<8>and1⊔1<5>⊔1<9>.4.5.5The case of bidegree(4,4)The classiﬁcation of the real schemes of curves of bidegree(4,4)[16]shows that there is no congruence like4.5.1for curves of even degree:the Euler characteristic of surfaces B1 and B2for curves of bidegree(4,4)can be any even number between-8and10.Thus the condition that Dw2(C B)+[R B]+[C A]=0is essential.4.5.6Addendum,the Fiedler congruence for curves on an ellipsoidLet A be an M-curve of bidegree(d,d)on ellipsoid B.Suppose that d is even,the Euler characteristic of each component of B1is even andχ(B1)≡2(mod4)thenχ(B2)≡d2(mod16)χ(B1)≡2−d2(mod16)Proof。

Empirical processes of dependent random variables

2
Preliminaries
n i=1
from R to R. The centered G -indexed empirical process is given by (P n − P )g = 1 n
n
the marginal and empirical distribution functions. Let G be a class of measurabrocesses that have been discussed include linear processes and Gaussian processes; see Dehling and Taqqu (1989) and Cs¨ org˝ o and Mielniczuk (1996) for long and short-range dependent subordinated Gaussian processes and Ho and Hsing (1996) and Wu (2003a) for long-range dependent linear processes. A collection of recent results is presented in Dehling, Mikosch and Sorensen (2002). In that collection Dedecker and Louhichi (2002) made an important generalization of Ossiander’s (1987) result. Here we investigate the empirical central limit problem for dependent random variables from another angle that avoids strong mixing conditions. In particular, we apply a martingale method and establish a weak convergence theory for stationary, causal processes. Our results are comparable with the theory for independent random variables in that the imposed moment conditions are optimal or almost optimal. We show that, if the process is short-range dependent in a certain sense, then the limiting behavior is similar to that of iid random variables in that the limiting distribution is a Gaussian process and the norming √ sequence is n. For long-range dependent linear processes, one needs to apply asymptotic √ expansions to obtain n-norming limit theorems (Section 6.2.2). The paper is structured as follows. In Section 2 we introduce some mathematical preliminaries necessary for the weak convergence theory and illustrate the essence of our approach. Two types of empirical central limit theorems are established. Empirical processes indexed by indicators of left half lines, absolutely continuous functions, and piecewise diﬀerentiable functions are discussed in Sections 3, 4 and 5 respectively. Applications to linear processes and iterated random functions are made in Section 6. Section 7 presents some integral and maximal inequalities that may be of independent interest. Some proofs are given in Sections 8 and 9.

Method and device for acquiring and displaying an

专利名称：Method and device for acquiring anddisplaying an immunofluorescence image ofa biological sample发明人：SUMPF, Tilman Johannes,PAPE,Michael,HAGEN-EGGERT, Martin,PIEPKE,Christian,LAUDAN, Thomas,FALKERT,Michael,MORRIN, Markus,RENTZSCH, Kristin 申请号：AU2019229421申请日：20190913公开号：AU2019229421A1公开日：20200402专利内容由知识产权出版社提供摘要：#$%^&*AU2019229421A120200402.pdf##### METHOD AND DEVICE FOR ACQUIRING AND DISPLAYING AN IMMUNOFLUORESCENCE IMAGE OF A BIOLOGICAL SAMPLE ABSTRACT A method is proposed for acquiring and displaying an immunofluorescence image of a biological sample (G). During a first operating state, the sample is continuously illuminated using the excitation radiation. A relative position between the sample (G) and an optical system, which guides the fluorescent radiation to an image sensor (BS), is changed in dependence on a movement gesture of a user and a corresponding digital image is displayed. Upon detection of a termination of the movement gesture, a change is made from the first operating state to a second operating state. In the second operating state, the sample is firstly illuminated using the excitation radiation to excite the fluorescent radiation of the fluorescent pigment. An acquisition of fluorescent radiation emitted by the sample (G) then takes place and acorresponding, further digital image is generated. In the second operating state, the acquisition of the emitted fluorescent radiation and the illumination of the sample are terminated after lapse of the second acquisition duration. Furthermore, the further digital image is continuously displayed after the termination of the acquisition of the fluorescent radiation and the termination of the illumination of the sample. 1/10 FIG.I1 BS, BS1 Ki K Sw FS KO j ST2 N -FS ES y7 scsc ISC7 F12 Fll -/Os :i-OB G STSS-N6 0 HS0 AE 申请人：EUROIMMUN Medizinische Labordiagnostika AG代理人：Spruson & Ferguson更多信息请下载全文后查看。

The University of Florida Sparse Matrix Collection

The University of Florida Sparse Matrix CollectionTimothy A.Davis1andYifan Hu2The University of Florida Sparse Matrix Collection is a large,widely available,and actively grow-ing set of sparse matrices that arise in real applications.Its matrices cover a wide spectrum of do-mains,include those arising from problems with underlying2D or3D geometry(such as structural engineering,computationalﬂuid dynamics,model reduction,electromagnetics,semiconductor de-vices,thermodynamics,materials,acoustics,computer graphics/vision,robotics/kinematics,and other discretizations)and those that typically do not have such geometry(such as optimization, circuit simulation,economic andﬁnancial modeling,theoretical and quantum chemistry,chemical process simulation,mathematics and statistics,power networks,and other networks and graphs). The collection is widely used by the sparse matrix algorithms community for the development and performance evaluation of sparse matrix algorithms.The collection includes software for accessing and managing the collection,from MATLAB,Fortran,and C,as well as online search capabil-ity.Graph visualization of the matrices is provided,and a new multilevel coarsening scheme is proposed to facilitate this task.Categories and Subject Descriptors:G.1.3[Numerical Analysis]:Numerical Linear Algebra—linear systems(direct methods),sparse and very large systems;G.4[Mathematics of Com-puting]:Mathematical Software—algorithm analysis,eﬃciency;G.2[Discrete Mathematics]: Graph TheoryGeneral Terms:Algorithms,Experimentation,PerformanceAdditional Key Words and Phrases:sparse matrices,performance evaluation,graph drawing, multilevel algorithms1.INTRODUCTIONAlthough James Wilkinson’s foundational work in numerical analysis touched only lightly upon sparse matrix computations,his deﬁnition of a sparse matrix is widely used([Gilbert et al.1992],for example).Wilkinson deﬁned a matrix as“sparse”if it has enough zeros that it pays to take advantage of them.He actually stated his deﬁnition in the negation:1Dept.of Computer and Information Science and Engineering,Univ.of Florida,Gainesville,FL, USA.davis@cise.uﬂ.edu.http://www.cise.uﬂ.edu/∼davis.Portions of this work were supported by the National Science Foundation,under grants9111263,9223088,9504974,9803599,0203720, and0620286.2AT&T Labs Research,Florham Park,NJ,USA.yifanhu@.Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for proﬁt or commercial advantage,the ACM copyright/server notice,the title of the publication,and its date appear,and notice is given that copying is by permission of the ACM,Inc.To copy otherwise,to republish, to post on servers,or to redistribute to lists requires prior speciﬁc permission and/or a fee.c 20YY ACM0098-3500/20YY/1200-0001$5.00ACM Transactions on Mathematical Software,Vol.V,No.N,M20YY,Pages1–0??.2·T.A.Davis and Y.HuThe matrix may be sparse,either with the non-zero elements concen-trated on a narrow band centered on the diagonal or alternatively theymay be distributed in a less systematic manner.We shall refer to amatrix as dense if the percentage of zero elements or its distributionis such as to make it uneconomic to take advantage of their presence.[Wilkinson1971]In other words,if you can save time or memory(usually both)by exploiting the zeros,then the matrix is sparse.An interesting aspect of this deﬁnition is that it is dependent not only on the matrix,but on the algorithm used on the matrix.For example,a matrix may be“sparse”for an iterative method for solving linear systems or for a graph theoretic algorithm,but not for a sparse factorization method.This article describes the University of Florida Sparse Matrix Collection(here-after referred to as the UF Collection),which contains sparse matrices arising in a wide range of applications.Section2gives the motivation for collecting sparse matrices from real applications and making them widely available.Section3de-scribes the current state of the collection and the breadth of problems the matrices represent.Section4describes the algorithm used,and new techniques developed, for visualizing the matrices.Section5describes the three data formats used for storing the matrices,and the kinds of auxiliary information available for each ma-trix.Section6describes the four methods for searching and downloading matrices of interest:a MATLAB interface(UFget),a Java interface(UFgui),the matrix web pages(via a standard browser and a web-based search tool),and Amazon Web Services TM.Examples of how the UF Collection can be used for performance eval-uation are given in Section7.The future of the UF Collection depends critically upon the submission of new matrices,as discussed in Section8.Finally,Section9 summarizes the contributions of the UF Collection and its associated software.In this paper,a graph or mesh is said to have2D or3D geometry if its vertices have a position on an xy or xyz plane that naturally arises from the problem being solved.A sparse matrix is said to have2D or3D geometry if its nonzero pattern is the adjacency matrix of such a graph.The notation|A|refers to the number of nonzeros in a matrix.2.MOTIVATIONThe role of sparse matrices from real applications in the development,testing, and performance evaluation of sparse matrix algorithms has long been recognized. Theﬁrst established collection was started by Duﬀand Reid(1970to1979,[Duﬀand Reid1979]),and then compiled into the Harwell-Boeing collection by Duﬀ, Grimes,and Lewis[Duﬀet al.1989].This collection provided the starting point of University of Florida Sparse Matrix Collection.Since the start of our collection in 1992,additional matrices have been added over the years,and other collections have been made,many of which have also been incorporated into the UF Collection(such as[Bai et al.1996;2008;Batagelj and Mrvar2008;Boisvert et al.2008;Boisvert et al.1997;Dumas2008;Gay2008;Gould et al.2008;Koster2008;M´e sz´a ros2008; Mittelmann2008;Resende et al.1995;Rudnyi2008;Rudnyi et al.2006;Saad2008; Schenk2008]).ACM Transactions on Mathematical Software,Vol.V,No.N,M20YY.University of Florida sparse matrix collection·3 The Matrix Market[Boisvert et al.1997]is the most similar collection to the UF Collection.Both collections include a search tool,and both categorize the matrices by application domain and problem source.Both provide matrices in similarﬁle formats.Both provide a web page for each matrix,with basic statistics andﬁgures. They diﬀer in size,with the UF Collection containing much larger matrices and 4.5times as many matrices.The latest matrix added to the Matrix Market was in 2000,whereas UF collection is constantly being updated with new matrices.The largest matrix in the Matrix Market has dimension90,449with2.5million nonzeros, whereas the largest matrix in the UF Collection has a dimension of28million with 760million nonzeros.Nearly every matrix in the Matrix Market is also included in the UF Collection.However,the Matrix Market does include matrix generators; the UF Collection has no matrix generators.Nearly all research articles that include a section on the performance analysis of a sparse matrix algorithm include results on matrices from real applications or parametrized matrices that mimic those that could arise in practice.Since maintaining a large collection of matrices from real applications is not trivial,an alternative isﬁrst considered,namely,parametrized and random matrix generators.2.1Parameterized and random matricesRandomized or repeatable parametrized sequences of matrices can be easily con-structed via simple matrix generators.Examples are listed below,as a comparison and contrast with the UF Collection,which does not include any matrix generators.(1)Simple discretizations of the Laplace operator on square2D and3D meshes.(2)The L-shaped meshes of[George and Liu1981].(3)Least-squares problems from square2Dﬁnite-element meshes[George et al.1983].(4)The LAPACK test matrix generators[Anderson et al.1999],which can generatebanded matrices and sparse matrices with random nonzero pattern.These also appear in the Matrix Market.(5)The Non-Hermitian Eigenvalue Problem(NEP)matrix generators[Bai et al.1996;2008].These include regular2D meshes and random patterns;additional matrices are made available only as operators(y=Ax,where A is not explicitly represented).(6)Zlatev’s matrix generators,which create a sparse Toeplitz structure(whereselected diagonals are all nonzero)[Zlatev1991].(7)Higham’s Matrix Computational Toolbox[Higham2002],part of which appearsas the gallery function in MATLAB.The gallery contains three functions that generate parametrized sparse matrices from regular2D meshes(neumann, poisson,and wathen)and two that generate banded matrices(toeppen and tridiag).No randomization is used,except in the values(but not pattern) of the wathen matrices.Many of these matrix generators also appear in the Matrix Market.(8)Matrices with purely random nonzero patterns[Erd¨o s and R´e nyi1959;Gilbert1959].They can be generated by sprand,sprand,and sprandsym in MATLAB [Gilbert et al.1992].ACM Transactions on Mathematical Software,Vol.V,No.N,M20YY.4·T.A.Davis and Y.Hu(9)The YM11subroutine in HSL can generate random sparse matrices,with op-tions for ensuring structural nonsingularity and bandedness(a matrix is struc-turally nonsingular if there exists a permutation so that the matrix has a zero-free diagonal)[Duﬀ2001].(10)In between the purely-random and purely-parametrized classes of matricesare partially randomized matrices with speciﬁc structure,as exempliﬁed by the CONTEST toolbox[Taylor and Higham2009].They provide a set of graph generators,some of which use an underlying2D or3D geometry and some that do not.For example,they include an implementation of the small-world graphs of[Kleinberg2000],which are2D meshes with additional randomized edges to distant vertices in the work generators in CONTEST that do not have geometry include the scale-free graph models of[Barab´a si and Albert1999].The prime advantage of random and parametrized matrices is that they are very easy to generate in any size desired.Matrices from real applications are very diﬃcult to generate and it is hard to vary them in size.Another key advantage of random and parametrized matrices is that asymptotic results can sometimes be derived.For example,the nested dissection ordering applied to a2D s-by-s mesh leads to an asymptotically optimal ordering for sparse Cholesky factorization,with31(n log2n)/8+O(n)nonzeros in the Cholesky factor L,and requiring829(n3/2)/84+O(n log n)ﬂoating point operations to compute, where n=s2is the dimension of the matrix[George and Liu1981].Purely random matrices may be useful for testing some sparse matrix applica-tions.For example,they can be useful for testing the convergence of iterative methods,assuming the generator has some control over the eigenvalue spectra(as is the case for sprandsym,for example).However,under Wilkinson’s deﬁnition they are not truly sparse when factorized by direct methods.With modest assumptions, purely random n-by-n matrices with O(n)nonzero entries require O(n3)time and O(n2)memory to factorize,because of catastrophicﬁll-in[Duﬀ1974].Catastrophic ﬁll-in very rarely occurs in matrices arising in real applications,and even when it does it is an indication that direct factorization methods are not applicable to that problem.Thus,performance obtained on purely random matrices will not indicate how well a sparse matrix factorization method will work on matrices from real ap-plications.Purely random networks also do not accurately model the characteristics of real networks[Watts and Strogatz1998].In between the two extremes of highly structured matrices(banded,or square 2D and3D meshes,for example)and purely randomized matrices is another class of matrix generators that can create matrices with some regular structure and a carefully selected random structure.The small-world graphs from the CONTEST toolbox are one such example.These are constructed to capture essential properties of large networks,such as the small-world property,scale-free degree distribution, motifs,and graphlet frequency[Taylor and Higham2009].These models are very useful for network algorithms,but they have not yet been shown to mimic the per-formance of sparse matrix factorization methods on matrices from real applications. ACM Transactions on Mathematical Software,Vol.V,No.N,M20YY.University of Florida sparse matrix collection·5Fig.1.Sparse matrix pattern of a RAH-66Comanche helicopter2.2Sparse matrices from real applicationsWhile useful,random and parametrized matrices have their limitations.This mo-tivates the development of the UF Collection,which focuses on matrices from real applications1.One of the matrices in the UF Collection is a complete representation of a Boe-ing/Sikorsky RAH-66Comanche helicopter,with3D geometry(obtained from Alex Pothen).Figure1shows the sparsity pattern of the matrix.Nested dissection is still useful for this problem,but asymptotic results cannot be derived for a complex structure such as this one.A simple square3D mesh would not be a good approxi-mation to this graph,since it would not capture the properties of the long helicopter rotors,or the topology of the hole where the rear rotor is located.This matrix is one of the few in the collection where the3D coordinates of the vertices are given;a picture of the graph drawing when these coordinates are used is shown in Figure2, on the left.For comparison,a force-directed graph drawing[Hu2005],which recre-ates these coordinates based only on the connectivity of the graph,is shown on the right.The method for creating thisﬁgure is discussed in Section4.The structure is warped,with the tail rotor twisted out of its housing(to the right)and itsﬁve main rotors shrunken(on the top),but it is still clear that the graph represents some kind of3D problem.Edge colors in the force-directed graphs represent the distance between two vertices.Figure3is the nonzero pattern of matrix arising from circuit simulation(left)and its force-directed graph drawing(right).The graphs of the Comanche helicopter and this electronic circuit indicate that one is a3D problem and the other is a network with no2D or3D geometry.This diﬀerence is not clear by merely comparing the nonzero patterns of their matrices.Parameterized matrices can capture many key properties of matrices from real applications,such as the regularity of a3D mesh,or the small-world properties of a network,but they cannot capture the rich and complex structure of matrices such1The UF Collection does include a handful of random matrices,which remain in the collection for historical reasons.ACM Transactions on Mathematical Software,Vol.V,No.N,M20YY.6·T.A.Davis and Y.HuFig.2.Graph of a RAH-66Comanche helicopter,using given3D coordinates(left)and its force-directed graph drawing(right)Fig.3.Sparse matrix pattern of an electronic circuit from Steve Hamm,Motorola(left)and its force-directed graph(right)as those presented above.2.3Collecting the matricesThe ideal matrix collection would be informally representative of matrices that arise in practice.It should cast a broad net so as to capture matrices from every applica-tion domain that relies on sparse matrix methods.For example,matrices arising in circuit simulation(and other network-related domains)diﬀer greatly from matrices arising from the discretization of2D and3D physical domains;this can be clearly seen in the precedingﬁputationalﬂuid dynamics matrices diﬀer from structural engineering matrices,and both are vastly diﬀerent from matrices arising in linear programming orﬁnancial portfolio optimization.The collection should be kept up to date,since matrices of interest grow in size each year as computer mem-ories get larger.New application domains also appear,such as eigenvalue problems arising in web connectivity matrices[Kamvar2008;Kamvar et al.2004;Page et al. 1998],which have existed only since the mid1990’s.ACM Transactions on Mathematical Software,Vol.V,No.N,M20YY.University of Florida sparse matrix collection·7 Sparse matrix algorithm developers use the matrices in the UF Collection to develop their methods,since theoretical asymptotic time/memory complexity anal-ysis only goes so far.If there are no matrices available to developers from a given application domain,it is quite possible that when their methods are used in that domain,the performance results will be disappointing.This provides a strong mo-tivation for computational scientists to submit their matrices to a widely available collection such as this one,so that gaps can be avoided.Thus,new application areas are always being added to the collection.Our strategy for adding matrices to the collection is simple,although admittedly ad hoc.Theﬁrst author maintains a collection of codes for the direct solution of sparse linear systems.End-users of this software are uniformly requested to submit matrices to the collection.Additional matrices are requested when they are found cited in articles and conference presentations,which includes a wider range of matrices(such as graphs arising in network analysis).Any matrices received are included,unless they are clear duplications of matrices already in the collection. Small matrices are not included,unless they are subsets of a larger collection(such as the Pajek data set).This strategy does introduce an unavoidable source of bias,but we have at-tempted to avoid this bias by relying on matrices collected by others.The UF Collection includes many sets of matrices collected in this way,such as those col-lected by Saad,who develops iterative methods for sparse linear systems[Saad 2003].Not all matrices in the collection arise from a sparse linear system.For example,many linear programming problems have been included in the UF Collec-tion[Gay2008;M´e sz´a ros2008;Mittelmann2008;Resende et al.1995].Network and combinatorial problems are also included,such as a matrix of prime numbers from Problem7of Trefethen’s100-digit challenge[Trefethen2002].Once a matrix is collected,its maintenance is the responsibility of theﬁrst author. This guards against arbitrary modiﬁcations in the matrices,which can occur when the matrix submitters modify their matrix problems.Repeatability of experiments based on the UF Collection is thus ensured.3.DESCRIPTION OF THE COLLECTIONAs of March2010the UF Sparse Matrix Collection consists of2272problems,each of which contains at least one sparse matrix(typically just one).It often represents a sparse linear system of the form Ax=b,where b is sometimes provided as well as the sparse matrix A.Many other problems are eigenvalue problems,and many matrices come from unstructured graphs and networks with no associated linear system.In some cases,a problem consists of a sequence of closely related matrices, such as Jacobian matrices arising in the solution of a nonlinear system.Some problems include two sparse matrices,such as a stiﬀness matrix and a mass matrix from a structural engineering eigenvalue problem.In this case,A is the stiﬀness matrix,and the mass matrix appears as an auxiliary matrix in the problem.3.1Application areasThe collection is divided into157diﬀerent matrix groups,with more groups added as new matrices are submitted to the collection.A complete list of these groups is too long to include here;details are given on the collection’s web site.Each groupACM Transactions on Mathematical Software,Vol.V,No.N,M20YY.8·T.A.Davis and Y.HuTable I.Partial list of sources of matricesNon-Hermitian Eigenvalue Problems[Bai et al.1996]Pajek Networks[Batagelj and Mrvar2008]Multistage stochasticﬁnancial modeling[Berger et al.1995]The Matrix Market(collection)[Boisvert et al.1997]Univ.of Utrecht circuit simulation[Bomhof and van der Vorst2000]MRI reconstruction M.Bydder,UCSDHarwell-Boeing Collection[Duﬀet al.1989]Combinatorial problems[Dumas2008]Frequency domain,nonlinear analog circuits[Feldmann et al.1996]NETLIB Linear Programming Test Problems[Gay2008]Symmetric sparse matrix benchmarks[Gould et al.2008]Linear programming problems[Gupta1996]Stanford/Berkeley Web Matrices[Kamvar et al.2004]Xyce circuit simulation matrices[Keiter et al.2003]Computer vision problems[Kemelmacher2005]PARASOL Matrix Collection[Koster2008]Linear programming test set[M´e sz´a ros2008]2D and3D semiconductor physics[Miller and Wang1991]Linear programming test set[Mittelmann2008]QAPLIB,quadratic assignment[Resende et al.1995]Oberwolfach Model Reduction Benchmarks[Rudnyi et al.2006]SPARSKIT matrix collection[Saad2008]Univ.of Basel Collection[Schenk2008]DNA electrophoresis matrices[van Heukelum et al.2002]Chemical engineering problems[Zitney1992;Zitney et al.1996]typically consists of a set of matrices from a single source.A few of the matrix groups in the UF Collection consist of entire sets of matrices from another sparse matrix collection.In this case,the group may consist of matrices from very diﬀerent application areas.For example,the Harwell-Boeing collection forms a single group [Duﬀand Reid1979;Duﬀet al.1989].Sources(papers and web sites)for some of the matrix groups are listed in Table I,which gives aﬂavor of the range of problems the collection contains.The group of a problem forms part of the full name of the problem.For exam-ple,the full name of the west0479problem in the Harwell-Boeing collection(the HB group in the UF sparse matrix collection)is HB/west0479.In addition,all problems are tagged with a string,kind,which indicates the application domain of the prob-lem.The group and kind of a problem are not the same.Some groups have many diﬀerent kinds of problems,and problems of the same kind can appear in diﬀerent groups.For example,the HB/west0479problem is in the HB group,and its kind is tagged as a“chemical process simulation problem.”Five other groups include chemical process simulation problems,namely,the Bai,Grund,Mallya,VanVelzen, and Zitney groups,all of which are named after the person from whom the matrices were obtained.A complete list of the kind strings for all problems is given in Table II.The table is split into two categories:problems with no underlying geometry and problems with2D or3D geometry.The collection contains68matrices with random nonzero pattern.They appear in the collection only because they already occur as widely-used test problems in ACM Transactions on Mathematical Software,Vol.V,No.N,M20YY.University of Florida sparse matrix collection·9 Table II.Summary of Problem.kind for all2272problems1516problems with no2D/3D geometry70chemical process simulation problem251circuit simulation problem299combinatorial problem11counter-example problem68economic problem4frequency-domain circuit simulation problem23least squares problem342linear programming problem135optimization problem56power network problem10statistical/mathematical problem61theoretical/quantum chemistry problem88directed graph8bipartite graph23undirected graph68random graph756problems with2D/3D geometry13acoustics problem166computationalﬂuid dynamics problem12computer graphics/vision problem44electromagnetics problem28materials problem42model reduction problem3robotics problem35semiconductor device problem288structural problem31thermal problem942D/3D problem(other than those listed above)another collection that was subsequently added to the UF sparse matrix collection. In retrospect,their inclusion in the UF Collection was a mistake,but matrices are never removed once added,since this would cause confusion when replicating results that use the collection.We guarantee statements such as“all matrices with property X in the collection as of date Y”always give a single consistent answer.3.2Matrix statisticsThe2272matrices in the collection come from359diﬀerent authors and50diﬀerent editors.A matrix editor is the one who collected the matrix for inclusion into any established collection(not just the UF Collection2,but also for others such as the Harwell/Boeing collection3,the Matrix Market4,or the GRID/TSLE collection5).A matrix author is the one who created the matrix.Each matrix has an editor and an author(sometimes the same person).Editors contributing at least10matrices are listed in Table III(sorted by the number of matrices collected).Figures4and5plot the matrix size(dimension and number of nonzeros)versus 2UF Sparse Matrix Collection:http://www.cise.uﬂ.edu/research/sparse/matrices3Harwell/Boeing collection:/nag/hb/hb.shtml4Matrix Market:/MatrixMarket5GRID/TLSE collection:http://gridtlse.enseeiht.fr:8080/websolve/ACM Transactions on Mathematical Software,Vol.V,No.N,M20YY.10·T.A.Davis and Y.HuFig.4.Matrix dimension(the largest of row/column dimension if rectangular)versus year created. The solid line is the cumulative sum.Fig.5.Number of nonzeros in each matrix versus year created.The solid line is the cumulative sum.ACM Transactions on Mathematical Software,Vol.V,No.N,M20YY.University of Florida sparse matrix collection·11 Table III.Primary editors of the UF Collection#matrices editor720T.Davis293J.-G.Dumas255I.Duﬀ,R.Grimes,J.Lewis166 C.Meszaros103O.Schenk78Z.Bai,D.Day,J.Demmel,J.Dongarra72V.Batagelj67Y.Ye61 A.Baggag,Y.Saad48 D.Gay43R.Fourer39N.Gould,Y.Hu,J.Scott38 E.Rudnyi35G.Kumfert,A.Pothen34 A.Curtis,I.Duﬀ,J.Reid28J.Chinneck27N.Gould25H.Mittelmann23R.Boisvert,R.Pozo,K.Remington,ler,R.Lipman,R.Barrett,J.Dongarra23 F.Grund22J.Koster13I.Lustig12H.Simon11M.ResendeFig.6.Overall histograms of matrix dimensions and nonzerosACM Transactions on Mathematical Software,Vol.V,No.N,M20YY.12·T.A.Davis and Y.Huthe year in which the matrices were created.The solid line in theﬁgures is the cumulative sum of the data plotted.Bothﬁgures show an exponential growth in problem sizes,similar to how computer memory sizes have grown since1970. Note that small problems are still being added to the collection.This is because a matrix group often includes a range of related problems,from small test cases to the largest problems of interest.The outlier matrix in1971is from the Edinburgh Associative Thesaurus,located at and obtained from the Pajek data set[Batagelj and Mrvar 2008].It is a graph with23,219vertices and325,592edges that wasﬁrst constructed in1971[Kiss et al.1973].Figure6plots two histograms of the overall distribution of matrices in the col-lection.4.VISUALIZING THE COLLECTIONMany basic facts,including symmetry,structural rank,ordering statistics,as well as a plot of the sparsity pattern,are given for each matrix in the collection.However these facts alone do not always give suﬃcient information about the matrix.For example,does this matrix come from an application involving2D or3D mesh?Or from a small-world network?Are there other structures in the applications that are not discernible from a plot of the sparsity pattern?To help answer these questions,a visualization of each matrix in the form of graph drawing is provided.If the matrix is structurally symmetric,it is taken as the adjacency matrix of an undirected graph,where two vertices i and j are connected if the(i,j)-th entry of the matrix is nonzero.Rectangular or structurally unsymmetric matrices are treated as bipartite graphs.More speciﬁcally,the augmented matrix0A A T0is used as the adjacency matrix of an undirected bipartite graph whose vertices are composed of the rows and columns of the matrix.We provide two graphs for square matrices with unsymmetric structure:the graph of A+A T and the augmented matrix above.The graph drawings depend only on the nonzero pattern,not the values.The basic algorithm used for drawing the graphs is a multilevel force-directed algorithm[Hu2005].However this algorithm fails to produce informative drawings for some matrices.We propose here a new coarsening scheme to deal with these cases.In the following we brieﬂy describe the basic algorithm,followed by the new coarsening scheme.4.1The graph drawing algorithmThe basic algorithm[Hu2005]employs a force-directed model[Fruchterman and Reingold1991].Vertices are treated as particles with electrical charges that push each other away.This repulsive force F r(i,j)between any two vertices i and j is proportional to the inverse of their distance,F r(i,j)=K2x i−x j ,i=j.ACM Transactions on Mathematical Software,Vol.V,No.N,M20YY.。

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Some properties of the Toda system: Lax pair and spinors . . . . . . . . . .
1
6 Conclusions A Atiyah-Hitchin metric A.1 The solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Global considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Asymptotic behaviour of the Atiyah-Hitchin metric . . . . . . . . . . . . . .
35 36 36 38 40
2
1
Introduction
Metrics of special holonomy have played an important role in many areas of string theory and M-theory. Compact metrics have been used extensively for Kaluza-Klein dimensional reduction to four, and other dimensions. Non-compact metrics of special holonomy have been used as generalisations of the usual ﬂat metric on the space transverse to a p-brane, providing conﬁgurations with lesser supersymmetry, and by turning on additional ﬂuxes, conﬁgurations that break other symmetries, such as the conformal symmetry in the usual AdS/CFT correspondence. In this paper, we shall investigate several examples involving new metrics with special holonomy. Our ﬁrst example is a family of eight-dimensional metrics of cohomogeneity one and Spin(7) holonomy, deﬁned on a certain complex line bundle over CP3 . The principal orbits in these metrics are S 7 , viewed as a triaxially-squashed S 3 bundle over S 4 . The system of four ﬁrst-order equations following from requiring Spin(7) holonomy were obtained recently in [1, 2], and a numerical analysis in [2] indicated that complete asymptotically local conical (ALC) metrics, denoted by C8 there, arise as solutions of these equations. There is a nontrivial parameter, characterising the degree of squashing of the CP3 bolt. These solutions were found to have a behaviour similar to that seen in the four-dimensional Atiyah-Hitchin metric [3, 4], with one direction on the S 3 ﬁbres collapsing on the singular orbit at short distance, whilst a diﬀerent direction in the S 3 stabilises to a constant radius at large distance [2]. If one takes the product of the Atiyah-Hitchin metric and seven-dimensional Minkowski spacetime, one obtains a solution of D = 11 supergravity which, after reduction on the circle of asymptotically-stabilised radius, admits an interpretation as a D6-brane orientifold plane [5]. We show in this paper that an analogous interpretation can be given to an M-theory solution of the product of a C8 metric with 3-dimensional Minkowski spacetime, which from the type IIA viewpoint becomes D6-brane orientifold plane wrapped on S 4 . We do this by making a detailed analysis of the coordinate identiﬁcations required by the regularity of the C8 metrics, and by making a perturbative analysis of the large-distance asymptotics, to establish the mass of the solutions. Our second example is concerned with seven-dimensional metrics of cohomogeneity one [2, 6], with six metric functions characterising homogeneous deformations of the S 3 × S 3 , and G2 holonomy. A wide class of metrics with S 3 × S 3 principal orbits was studied in
3 New G2 metrics C7 with S 3 × S 3 principal orbits 3.1 3.2 Analysis for C7 metrics with six-function solutions . . . . . . . . . . . . . . Analysis for C7 metrics for the four-function truncation . . . . . . . . . . . 3.2.1 3.2.2 3.2.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global structure of C7 metrics . . . . . . . . . . . . . . . . . . . . . Reduction to a type IIA solution . . . . . . . . . . . . . . . . . . . .
5 The Toda equation 5.1 5.2 Perturbation around the n = 4 ﬂat metric . . . . . . . . . . . . . . . . . . . Metrics with both self-dual and anti-self-dual Killing vectors . . . . . . . . . 5.2.1 5.2.2 5.2.3 5.2.4 5.3 The n = 0 Cherkis-Kapustin metric . . . . . . . . . . . . . . . . . . The Calderbank-Tod-Nutku-Sheftel solution . . . . . . . . . . . . . . Separable solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
DAMTP-2001-101
CTP TAMU-P-01-55
November 2001
hep-th/0111096
Orientifolds and Slumps in G2 and Spin(7) Metrics
M. Cvetiˇ c † , G.W. Gibbons ♯, H. L¨ u ⋆ and C.N. Pope ‡
We discuss some new metrics of special holonomy, and their roles in string theory and M-theory. First we consider Spin(7) metrics denoted by C8 , which are complete on a complex line bundle over CP3 . The principal orbits are S 7 , described as a triaxially squashed S 3 bundle over S 4 . The behaviour in the S 3 directions is similar to that in the AtiyahHitchin metric, and we show how this leads to an M-theory interpretation with orientifold D6-branes wrapped over S 4 . We then consider new G2 metrics which we denote by C7 , which are complete on an R2 bundle over T 1,1 , with principal orbits that are S 3 × S 3 . We study the C7 metrics using numerical methods, and we ﬁnd that they have the remarkable property of admitting a U (1) Killing vector whose length is nowhere zero or inﬁnite. This allows one to make an everywhere non-singular reduction of an M-theory solution to give a solution of the type IIA theory. The solution has two non-trivial S 2 cycles, and both carry magnetic charge with respect to the R-R vector ﬁeld. We also discuss some four-dimensional hyper-K¨ ahler metrics described recently by Cherkis and Kapustin, following earlier work by Kronheimer. We show that in certain cases these metrics, whose explicit form is known only asymptotically, can be related to metrics characterised by solutions of the su(∞) Toda equation, which can provide a way of studying their interior structure.