$*$-双单型$A \omega^2$-半群的同构定理

在本文中,我们研究了满足条件$D^*=D(~)$的*-双单型$A\omega^2$-半群的同构,给出了它们之间的同构定理.     

Wiener-Hopf operators on L_{\omega}^{2}(\mathbb{R}^{+})

Let be a weighted space with weight . In this paper we show that for every Wiener-Hopf operator T on and for every a I, there exists a function such that

Spectral theorem for multipliers on {L_{\omega}^{2}(\mathbb{R})}

We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{\omega}^{2}(\mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces L_w²(\mathbbR){L_{\omega}^{2}(\mathbb{R})} For operators M in the algebra generated by the convolutions with f ? C_c(\mathbb R){\phi \in {C_c(\mathbb {R})}} we show that [`(m(W))] = s(M){\overline{\mu(\Omega)} = \sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{\overline{\mu(\Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces L²_w(\mathbb R^k){L^2_{\omega}(\mathbb {R}^{k})} where the weight ω has a special form.     

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      5. Ideals in MOD-R and the {omega}-radical      Prest  Mike 《Journal London Mathematical Society》2005,71(2):321-334 Let R be an artin algebra, and let mod-R denote the categoryof finitely presented right R-modules. The radical rad = rad(mod-R)of this category and its finite powers play a major role inthe representation theory of R. The intersection of these finitepowers is denoted rad, and the nilpotence of this ideal hasbeen investigated, in [6, 13] for instance. In [17], arbitrarytransfinite powers, rad, of rad were defined and linked to theextent to which morphisms in mod-R may be factorised. In particular,it has been shown that if R is an artin algebra, then the transfiniteradical, rad, the intersection of all ordinal powers of rad,is non-zero if and only if there is a ‘factorisable system’of morphisms in rad and, in that case, the Krull–Gabrieldimension of mod-R equals (that is, is undefined). More preciseresults on the index of nilpotence of rad for artin algebraswere proved in [14, 20, 24–26].      6. Decompositions of Singular Continuous Spectra of \mathcal{H}_{ - 2} -class Rank One Perturbations      Sergio Albeverio  Alexei Konstantinov  Volodymyr Koshmanenko 《Integral Equations and Operator Theory》2005,52(4):455-464 The decomposition theory for the singular continuous spectrum of rank one singular perturbations is studied. A generalization of the well-known Aronszajn-Donoghue theory to the case of decompositions with respect to α-dimensional Hausdorff measures is given and a characterization of the supports of the α-singular, α-absolutely continuous, and strongly α-continuous parts of the spectral measure of - class rank one singular perturbations is given in terms of the limiting behaviour of the regularized Borel transform.      7. The shift graph and the Ramsey degree of {[\mathbb{N}]}^{\omega}      Carlos A. Di Prisco  Stevo Todorcevic 《Acta Mathematica Hungarica》2014,142(2):484-493 We address several questions related to the Ramsey degree of the class of infinite structures isomorphic to the positive integers with the usual ordering. In particular we study the strength of stating that this class has finite Ramsey degree in the absence of the axiom of choice. We show that if the shift graph has finite chromatic number then this class has infinite Ramsey degree.      8. A New \mathcal{N}\mathcal{P} -Complete Problem and Public-Key Identification      David Pointcheval  Guillaume Poupard 《Designs, Codes and Cryptography》2003,28(1):5-31 The appearance of the theory of zero-knowledge, presented by Goldwasser, Micali and Rackoff in 1985, opened a way to secure identification schemes. The first application was the famous Fiat-Shamir scheme based on the problem of modular square roots extraction. In the following years, many other schemes have been proposed, some Fiat-Shamir extensions but also new discrete logarithm based schemes. Therefore, all of them were based on problems from number theory. Their main common drawback is high computational load because of arithmetical operations modulo large integers. Implementation on low-cost smart cards was made difficult and inefficient.With the Permuted Kernels Problem (PKP), Shamir proposed the first efficient scheme allowing for an implementation on such low-cost smart cards, but very few others have afterwards been suggested.In this paper, we present an efficient identification scheme based on a combinatorial -complete problem: the Permuted Perceptrons Problem (PPP). This problem seems hard enough to be unsolvable even with very small parameters, and some recent cryptanalysis studies confirm that position. Furthermore, it admits efficient zero-knowledge proofs of knowledge and so it is well-suited for cryptographic purposes. An actual implementation completes the optimistic opinion about efficiency and practicability on low-cost smart cards, and namely with less than 2KB of EEPROM and just 100 Bytes of RAM and 6.4 KB of communication.      9. Singular minimizers of strictly polyconvex functionals in {\bf R}^{2 \times 2}      Jonathan Bevan 《Calculus of Variations and Partial Differential Equations》2005,23(3):347-372 We give examples of non-C1 one-homogeneous mappings and non-negative strictly polyconvex functions f such that where B denotes the unit ball in . Such u are therefore singular minimizers of the corresponding strictly polyconvex functionals in appropriate Sobolev spaces. Received: 12 January 2004, Accepted: 13 September 2004, Published online: 10 December 2004      10. Automorphisms of Relatively Free Groups in the Variety N2A {wedge} AN2 {wedge} Nc      Papistas  Athanassios I. 《Journal London Mathematical Society》2001,63(3):607-622 For positive integers n and c, with n 2, let Gn, c be a relativelyfree group of finite rank n in the variety N2A AN2 Nc. Itis shown that the subgroup of the automorphism group Aut(Gn,c) of Gn, c generated by the tame automorphisms and an explicitlydescribed finite set of IA-automorphisms of Gn, c has finiteindex in Aut(Gn, c). Furthermore, it is proved that there areno non-trivial elements of Gn, c fixed by every tame automorphismof Gn, c.      11. On the Diophantine equation $$X^{2N}+2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5$$      Eva G. Goedhart  Helen G. Grundman 《Periodica Mathematica Hungarica》2017,75(2):196-200 We prove that for each prime p, positive integer \(\alpha \), and non-negative integers \(\beta \) and \(\gamma \), the Diophantine equation \(X^{2N} + 2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5\) has no solution with N, X, \(Z\in \mathbb {Z}^+\), \(N > 1\), and \(\gcd (X,Z) = 1\).      12. Unique Ergodicity of Harmonic Currents On Singular Foliations of {\mathbb{P}^2}      John Erik Fornæss  Nessim Sibony 《Geometric And Functional Analysis》2010,19(5):1334-1377 Let F{\mathcal{F}} be a holomorphic foliation of \mathbbP2{\mathbb{P}^2} by Riemann surfaces. Assume all the singular points of F{\mathcal{F}} are hyperbolic. If F{\mathcal{F}} has no algebraic leaf, then there is a unique positive harmonic (1, 1) current T of mass one, directed by F{\mathcal{F}}. This implies strong ergodic properties for the foliation F{\mathcal{F}}. We also study the harmonic flow associated to the current T.      13. Isometric embeddings of \mathbb{Z}_{p^k} in the Hamming space \mathbb{F}_{p}^{N} and \mathbb{Z}_{p^k}-linear codes      Marcelo Muniz 《Designs, Codes and Cryptography》2006,41(2):147-152 Isometric embeddings of $\mathbb{Z}_{p^n+1}$ into the Hamming space ( $\mathbb{F}_{p}^{p^n},w$ ) have played a fundamental role in recent constructions of non-linear codes. The codes thus obtained are very good codes, but their rate is limited by the rate of the first-order generalized Reed–Muller code—hence, when n is not very small, these embeddings lead to the construction of low-rate codes. A natural question is whether there are embeddings with higher rates than the known ones. In this paper, we provide a partial answer to this question by establishing a lower bound on the order of a symmetry of ( $\mathbb{F}_{p}^{N},w$ ).      14. Classification of rational holomorphic maps from \mathbb{B}^2 into \mathbb{B}^\mathbb{N} with degree 2      ShanYu Ji  Yuan Zhang 《中国科学 数学(英文版)》2009,52(12):2647-2667 Rational proper holomorphic maps from the unit ball in ?2 into the unit ball ? N with degree 2 are classified, up to automorphisms of balls.      15. Dirac Operators with Singular Potentials Supported on Unbounded Surfaces in $$\mathbb{R}^{3}$$      Rabinovich  V. S. 《Functional Analysis and Its Applications》2021,55(3):245-249 Functional Analysis and Its Applications - We consider the self-adjointness and essential spectrum of 3D Dirac operators with bounded variable magnetic and electrostatic potentials and with...      16. Bispectral and (\mathfrak{gl}_{N},\mathfrak{gl}_{M}) dualities      Evgenii E. Mukhin  Vitaly O. Tarasov  Alexander N. Varchenko 《Functional Analysis and Other Mathematics》2006,1(1):47-69 Let be a space of quasipolynomials of dimension N=N 1+⋅⋅⋅+N n . We define the regularized fundamental operator of V as the polynomial differential operator D=∑ i=0 N A N−i (x)∂ x i annihilating V and such that its leading coefficient A 0 is a polynomial of the minimal possible degree. We apply a suitable integral transformation to V to construct a space of quasipolynomials whose regularized fundamental operator is the differential operator ∑ i=0 N u i A N−i (∂ u ). Our integral transformation corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity) of the KP hierarchy. As a corollary of the properties of the integral transformation, we obtain a correspondence between critical points of the two master functions associated with the -dual Gaudin models and also between the corresponding Bethe vectors. The research of E. M. was supported in part by the NSF (Grant No. DMS-0140460). The research of A. V. was supported in part by the NSF (Grant No. DMS-0244579).      17. Conservative fragments of {{S}^{1}_{2}} and {{R}^{1}_{2}}      Chris Pollett 《Archive for Mathematical Logic》2011,50(3-4):367-393 Conservative subtheories of ${{R}^{1}_{2}}$ and ${{S}^{1}_{2}}$ are presented. For ${{S}^{1}_{2}}$ , a slight tightening of Je?ábek??s result (Math Logic Q 52(6):613?C624, 2006) that ${T^{0}_{2} \preceq_{\forall \Sigma^{b}_{1}}S^{1}_{2}}$ is presented: It is shown that ${T^{0}_{2}}$ can be axiomatised as BASIC together with induction on sharply bounded formulas of one alternation. Within this ${\forall\Sigma^{b}_{1}}$ -theory, we define a ${\forall\Sigma^{b}_{0}}$ -theory, ${T^{-1}_{2}}$ , for the ${\forall\Sigma^{b}_{0}}$ -consequences of ${S^{1}_{2}}$ . We show ${T^{-1}_{2}}$ is weak by showing it cannot ${\Sigma^{b}_{0}}$ -define division by 3. We then consider what would be the analogous ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ based on Pollett (Ann Pure Appl Logic 100:189?C245, 1999. It is shown that this theory, ${{T}^{0,\left\{2^{(||\dot{id}||)}\right\}}_{2}}$ , also cannot ${\Sigma^{b}_{0}}$ -define division by 3. On the other hand, we show that ${{S}^{0}_{2}+open_{\{||id||\}}}$ -COMP is a ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ . Finally, we give a refinement of Johannsen and Pollett (Logic Colloquium?? 98, 262?C279, 2000) and show that ${\hat{C}^{0}_{2}}$ is ${\forall\hat\Sigma^{b}_{1}}$ -conservative over a theory based on open cl-comprehension.      18. Global bifurcation for quasilinear elliptic equations on $\mathbb{R}^{N}$      P.J. Rabier  C.A. Stuart 《Mathematische Zeitschrift》2001,237(1):85-124 In this paper we discuss the global behaviour of some connected sets of solutions of a broad class of second order quasilinear elliptic equations for where is a real parameter and the function u is required to satisfy the condition The basic tool is the degree for proper Fredholm maps of index zero in the form due to Fitzpatrick, Pejsachowicz and Rabier. To use this degree the problem must be expressed in the form where J is an interval, X and Y are Banach spaces and F is a map which is Fredholm and proper on closed bounded subsets. We use the usual spaces and . Then the main difficulty involves finding general conditions on and b which ensure the properness of F. Our approach to this is based on some recent work where, under the assumption that and b are asymptotically periodic in x as $\left| x\right| \rightarrow\infty$, we have obtained simple conditions which are necessary and sufficient for to be Fredholm and proper on closed bounded subsets of X. In particular, the nonexistence of nonzero solutions in X of the asymptotic problem plays a crucial role in this issue. Our results establish the bifurcation of global branches of solutions for the general problem. Various special cases are also discussed. Even for semilinear equations of the form our results cover situations outside the scope of other methods in the literature. Received March 30, 1999; in final form January 17, 2000 / Published online February 5, 2001      19. The Mesa Problem: Diffusion Patterns for ut = {nabla} {middle dot} (um{nabla}u) as m -> +{infty}      ELLIOTT  C. M.; HERRERO  M. A.; KING  J. R.; OCKENDON  J. R. 《IMA Journal of Applied Mathematics》1986,37(2):147-154 In this paper we consider the limit m+ of solutions of the porous-mediumequation ut = · (umu) (xRN), with N > 1. We conjecturethat, for initial data with a unique maximum, the evolutionis characterized by the onset of a ‘mesa’ region,in which the solution is nearly spatially independent, surroundedby a region in which u is nearly equal to its initial value.The transition between these regions occurs near a surface whichis identified with the free boundary in a certain Stefan problemwhich can be studied using variational inequalities. Moreover,singular-perturbation theory can be used to describe the structureof the transition region.      20. Navigating in the Cayley Graphs of {\rm SL}_{N}(\mathbb{Z}) and {\rm SL}_{N}(\mathbb{F}_{p})      T. R. Riley 《Geometriae Dedicata》2005,113(1):215-229 We give a nondeterministic algorithm that expresses elements of , for N ≥ 3, as words in a finite set of generators, with the length of these words at most a constant times the word metric. We show that the nondeterministic time-complexity of the subtractive version of Euclid’s algorithm for finding the greatest common divisor of N ≥ 3 integers a1, ..., aN is at most a constant times . This leads to an elementary proof that for N ≥ 3 the word metric in is biLipschitz equivalent to the logarithm of the matrix norm – an instance of a theorem of Mozes, Lubotzky and Raghunathan. And we show constructively that there exists K>0 such that for all N ≥ 3 and primes p, the diameter of the Cayley graph of with respect to the generating set is at most .Mathematics Subject Classification: 20F05

Ideals in MOD-R and the {omega}-radical

Let R be an artin algebra, and let mod-R denote the categoryof finitely presented right R-modules. The radical rad = rad(mod-R)of this category and its finite powers play a major role inthe representation theory of R. The intersection of these finitepowers is denoted rad, and the nilpotence of this ideal hasbeen investigated, in [6, 13] for instance. In [17], arbitrarytransfinite powers, rad, of rad were defined and linked to theextent to which morphisms in mod-R may be factorised. In particular,it has been shown that if R is an artin algebra, then the transfiniteradical, rad, the intersection of all ordinal powers of rad,is non-zero if and only if there is a ‘factorisable system’of morphisms in rad and, in that case, the Krull–Gabrieldimension of mod-R equals (that is, is undefined). More preciseresults on the index of nilpotence of rad for artin algebraswere proved in [14, 20, 24–26].     

Decompositions of Singular Continuous Spectra of \mathcal{H}_{ - 2} -class Rank One Perturbations

The decomposition theory for the singular continuous spectrum of rank one singular perturbations is studied. A generalization of the well-known Aronszajn-Donoghue theory to the case of decompositions with respect to α-dimensional Hausdorff measures is given and a characterization of the supports of the α-singular, α-absolutely continuous, and strongly α-continuous parts of the spectral measure of - class rank one singular perturbations is given in terms of the limiting behaviour of the regularized Borel transform.     

The shift graph and the Ramsey degree of {[\mathbb{N}]}^{\omega}

We address several questions related to the Ramsey degree of the class of infinite structures isomorphic to the positive integers with the usual ordering. In particular we study the strength of stating that this class has finite Ramsey degree in the absence of the axiom of choice. We show that if the shift graph has finite chromatic number then this class has infinite Ramsey degree.     

A New \mathcal{N}\mathcal{P} -Complete Problem and Public-Key Identification

The appearance of the theory of zero-knowledge, presented by Goldwasser, Micali and Rackoff in 1985, opened a way to secure identification schemes. The first application was the famous Fiat-Shamir scheme based on the problem of modular square roots extraction. In the following years, many other schemes have been proposed, some Fiat-Shamir extensions but also new discrete logarithm based schemes. Therefore, all of them were based on problems from number theory. Their main common drawback is high computational load because of arithmetical operations modulo large integers. Implementation on low-cost smart cards was made difficult and inefficient.With the Permuted Kernels Problem (PKP), Shamir proposed the first efficient scheme allowing for an implementation on such low-cost smart cards, but very few others have afterwards been suggested.In this paper, we present an efficient identification scheme based on a combinatorial -complete problem: the Permuted Perceptrons Problem (PPP). This problem seems hard enough to be unsolvable even with very small parameters, and some recent cryptanalysis studies confirm that position. Furthermore, it admits efficient zero-knowledge proofs of knowledge and so it is well-suited for cryptographic purposes. An actual implementation completes the optimistic opinion about efficiency and practicability on low-cost smart cards, and namely with less than 2KB of EEPROM and just 100 Bytes of RAM and 6.4 KB of communication.     

Singular minimizers of strictly polyconvex functionals in {\bf R}^{2 \times 2}

We give examples of non-C¹ one-homogeneous mappings and non-negative strictly polyconvex functions f such that where B denotes the unit ball in . Such u are therefore singular minimizers of the corresponding strictly polyconvex functionals in appropriate Sobolev spaces. Received: 12 January 2004, Accepted: 13 September 2004, Published online: 10 December 2004     

Automorphisms of Relatively Free Groups in the Variety N2A {wedge} AN2 {wedge} Nc

For positive integers n and c, with n 2, let G_{n, c} be a relativelyfree group of finite rank n in the variety N₂A AN₂ N_c. Itis shown that the subgroup of the automorphism group Aut(G_n,c) of G_{n, c} generated by the tame automorphisms and an explicitlydescribed finite set of IA-automorphisms of G_{n, c} has finiteindex in Aut(G_{n, c}). Furthermore, it is proved that there areno non-trivial elements of G_{n, c} fixed by every tame automorphismof G_{n, c}.     

On the Diophantine equation $$X^{2N}+2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5$$

We prove that for each prime p, positive integer \(\alpha \), and non-negative integers \(\beta \) and \(\gamma \), the Diophantine equation \(X^{2N} + 2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5\) has no solution with N, X, \(Z\in \mathbb {Z}^+\), \(N > 1\), and \(\gcd (X,Z) = 1\).     

Unique Ergodicity of Harmonic Currents On Singular Foliations of {\mathbb{P}^2}

Let F{\mathcal{F}} be a holomorphic foliation of \mathbbP²{\mathbb{P}^2} by Riemann surfaces. Assume all the singular points of F{\mathcal{F}} are hyperbolic. If F{\mathcal{F}} has no algebraic leaf, then there is a unique positive harmonic (1, 1) current T of mass one, directed by F{\mathcal{F}}. This implies strong ergodic properties for the foliation F{\mathcal{F}}. We also study the harmonic flow associated to the current T.     

Isometric embeddings of \mathbb{Z}_{p^k} in the Hamming space \mathbb{F}_{p}^{N} and \mathbb{Z}_{p^k}-linear codes

Isometric embeddings of $\mathbb{Z}_{p^n+1}$ into the Hamming space ( $\mathbb{F}_{p}^{p^n},w$ ) have played a fundamental role in recent constructions of non-linear codes. The codes thus obtained are very good codes, but their rate is limited by the rate of the first-order generalized Reed–Muller code—hence, when n is not very small, these embeddings lead to the construction of low-rate codes. A natural question is whether there are embeddings with higher rates than the known ones. In this paper, we provide a partial answer to this question by establishing a lower bound on the order of a symmetry of ( $\mathbb{F}_{p}^{N},w$ ).     

Classification of rational holomorphic maps from \mathbb{B}^2 into \mathbb{B}^\mathbb{N} with degree 2

Rational proper holomorphic maps from the unit ball in ?² into the unit ball ? ^N with degree 2 are classified, up to automorphisms of balls.     

Dirac Operators with Singular Potentials Supported on Unbounded Surfaces in $$\mathbb{R}^{3}$$

Functional Analysis and Its Applications - We consider the self-adjointness and essential spectrum of 3D Dirac operators with bounded variable magnetic and electrostatic potentials and with...     

Bispectral and (\mathfrak{gl}_{N},\mathfrak{gl}_{M}) dualities

Let be a space of quasipolynomials of dimension N=N ₁+⋅⋅⋅+N _n . We define the regularized fundamental operator of V as the polynomial differential operator D=∑ _i=0 ^N A _N−i (x)∂ _x ⁱ annihilating V and such that its leading coefficient A ₀ is a polynomial of the minimal possible degree. We apply a suitable integral transformation to V to construct a space of quasipolynomials whose regularized fundamental operator is the differential operator ∑ _i=0 ^N u ⁱ A _N−i (∂ _u ). Our integral transformation corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity) of the KP hierarchy. As a corollary of the properties of the integral transformation, we obtain a correspondence between critical points of the two master functions associated with the -dual Gaudin models and also between the corresponding Bethe vectors. The research of E. M. was supported in part by the NSF (Grant No. DMS-0140460). The research of A. V. was supported in part by the NSF (Grant No. DMS-0244579).     

Conservative fragments of {{S}^{1}_{2}} and {{R}^{1}_{2}}

Conservative subtheories of ${{R}^{1}_{2}}$ and ${{S}^{1}_{2}}$ are presented. For ${{S}^{1}_{2}}$ , a slight tightening of Je?ábek??s result (Math Logic Q 52(6):613?C624, 2006) that ${T^{0}_{2} \preceq_{\forall \Sigma^{b}_{1}}S^{1}_{2}}$ is presented: It is shown that ${T^{0}_{2}}$ can be axiomatised as BASIC together with induction on sharply bounded formulas of one alternation. Within this ${\forall\Sigma^{b}_{1}}$ -theory, we define a ${\forall\Sigma^{b}_{0}}$ -theory, ${T^{-1}_{2}}$ , for the ${\forall\Sigma^{b}_{0}}$ -consequences of ${S^{1}_{2}}$ . We show ${T^{-1}_{2}}$ is weak by showing it cannot ${\Sigma^{b}_{0}}$ -define division by 3. We then consider what would be the analogous ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ based on Pollett (Ann Pure Appl Logic 100:189?C245, 1999. It is shown that this theory, ${{T}^{0,\left\{2^{(||\dot{id}||)}\right\}}_{2}}$ , also cannot ${\Sigma^{b}_{0}}$ -define division by 3. On the other hand, we show that ${{S}^{0}_{2}+open_{\{||id||\}}}$ -COMP is a ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ . Finally, we give a refinement of Johannsen and Pollett (Logic Colloquium?? 98, 262?C279, 2000) and show that ${\hat{C}^{0}_{2}}$ is ${\forall\hat\Sigma^{b}_{1}}$ -conservative over a theory based on open _cl-comprehension.     

Global bifurcation for quasilinear elliptic equations on $\mathbb{R}^{N}$

In this paper we discuss the global behaviour of some connected sets of solutions of a broad class of second order quasilinear elliptic equations for where is a real parameter and the function u is required to satisfy the condition The basic tool is the degree for proper Fredholm maps of index zero in the form due to Fitzpatrick, Pejsachowicz and Rabier. To use this degree the problem must be expressed in the form where J is an interval, X and Y are Banach spaces and F is a map which is Fredholm and proper on closed bounded subsets. We use the usual spaces and . Then the main difficulty involves finding general conditions on and b which ensure the properness of F. Our approach to this is based on some recent work where, under the assumption that and b are asymptotically periodic in x as $\left| x\right| \rightarrow\infty$, we have obtained simple conditions which are necessary and sufficient for to be Fredholm and proper on closed bounded subsets of X. In particular, the nonexistence of nonzero solutions in X of the asymptotic problem plays a crucial role in this issue. Our results establish the bifurcation of global branches of solutions for the general problem. Various special cases are also discussed. Even for semilinear equations of the form our results cover situations outside the scope of other methods in the literature. Received March 30, 1999; in final form January 17, 2000 / Published online February 5, 2001     

The Mesa Problem: Diffusion Patterns for ut = {nabla} {middle dot} (um{nabla}u) as m -> +{infty}

In this paper we consider the limit m+ of solutions of the porous-mediumequation u_t = · (u^mu) (xR^N), with N > 1. We conjecturethat, for initial data with a unique maximum, the evolutionis characterized by the onset of a ‘mesa’ region,in which the solution is nearly spatially independent, surroundedby a region in which u is nearly equal to its initial value.The transition between these regions occurs near a surface whichis identified with the free boundary in a certain Stefan problemwhich can be studied using variational inequalities. Moreover,singular-perturbation theory can be used to describe the structureof the transition region.     

Navigating in the Cayley Graphs of {\rm SL}_{N}(\mathbb{Z}) and {\rm SL}_{N}(\mathbb{F}_{p})

We give a nondeterministic algorithm that expresses elements of , for N ≥ 3, as words in a finite set of generators, with the length of these words at most a constant times the word metric. We show that the nondeterministic time-complexity of the subtractive version of Euclid’s algorithm for finding the greatest common divisor of N ≥ 3 integers a₁, ..., a_N is at most a constant times . This leads to an elementary proof that for N ≥ 3 the word metric in is biLipschitz equivalent to the logarithm of the matrix norm – an instance of a theorem of Mozes, Lubotzky and Raghunathan. And we show constructively that there exists K>0 such that for all N ≥ 3 and primes p, the diameter of the Cayley graph of with respect to the generating set is at most .Mathematics Subject Classification: 20F05