text:
• ### Evaluation modules for quantum toroidal ${\mathfrak{gl}}_n$ algebras(1709.01592)

March 10, 2019 math-ph, math.MP, math.QA, math.RT
The affine evaluation map is a surjective homomorphism from the quantum toroidal ${\mathfrak {gl}}_n$ algebra ${\mathcal E}'_n(q_1,q_2,q_3)$ to the quantum affine algebra $U'_q\widehat{\mathfrak {gl}}_n$ at level $\kappa$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^n=\kappa^2$. We discuss ${\mathcal E}'_n(q_1,q_2,q_3)$ evaluation modules. We give highest weights of evaluation highest weight modules. We also obtain the decomposition of the evaluation Wakimoto module with respect to a Gelfand-Zeitlin type subalgebra of a completion of ${\mathcal E}'_n(q_1,q_2,q_3)$, which describes a deformation of the coset theory $\widehat{\mathfrak {gl}}_n/\widehat{\mathfrak {gl}}_{n-1}$.
• ### Self-dual Grassmannian, Wronski map, and representations of $\mathfrak{gl}_N$, ${\mathfrak{sp}}_{2r}$, ${\mathfrak{so}}_{2r+1}$(1705.02048)

March 27, 2018 math.AG, math.QA, math.RT
We define a $\mathfrak{gl}_N$-stratification of the Grassmannian of $N$ planes $\mathrm{Gr}(N,d)$. The $\mathfrak{gl}_N$-stratification consists of strata $\Omega_{\mathbf{\Lambda}}$ labeled by unordered sets $\mathbf{\Lambda}=(\lambda^{(1)},\dots,\lambda^{(n)})$ of nonzero partitions with at most $N$ parts, satisfying a condition depending on $d$, and such that $(\otimes_{i=1}^n V_{\lambda^{(i)}})^{\mathfrak{sl}_N}\ne 0$. Here $V_{\lambda^{(i)}}$ is the irreducible $\mathfrak{gl}_N$-module with highest weight $\lambda^{(i)}$. We show that the closure of a stratum $\Omega_{\mathbf{\Lambda}}$ is the union of the strata $\Omega_{\mathbf\Xi}$, $\mathbf{\Xi}=(\xi^{(1)},\dots,\xi^{(m)})$, such that there is a partition $\{I_1,\dots,I_m\}$ of $\{1,2,\dots,n\}$ with ${\rm {Hom}}_{\mathfrak{gl}_N} (V_{\xi^{(i)}}, \otimes_{j\in I_i}V_{\lambda^{(j)}}\big)\neq 0$ for $i=1,\dots,m$. The $\mathfrak{gl}_N$-stratification of the Grassmannian agrees with the Wronski map. We introduce and study the new object: the self-dual Grassmannian $\mathrm{sGr}(N,d)\subset \mathrm{Gr}(N,d)$. Our main result is a similar $\mathfrak{g}_N$-stratification of the self-dual Grassmannian governed by representation theory of the Lie algebra $\mathfrak {g}_{2r+1}:=\mathfrak{sp}_{2r}$ if $N=2r+1$ and of the Lie algebra $\mathfrak g_{2r}:=\mathfrak{so}_{2r+1}$ if $N=2r$.
• ### On the Gaudin model of type G$_2$(1711.02511)

March 13, 2018 math-ph, math.MP, math.AG, math.QA, math.RT
We derive a number of results related to the Gaudin model associated to the simple Lie algebra of type G$_2$. We compute explicit formulas for solutions of the Bethe ansatz equations associated to the tensor product of an arbitrary finite-dimensional irreducible module and the vector representation. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We show that the points of the spectrum of the Gaudin model in type G$_2$ are in a bijective correspondence with self-self-dual spaces of polynomials. We study the set of all self-self-dual spaces - the self-self-dual Grassmannian. We establish a stratification of the self-self-dual Grassmannian with the strata labeled by unordered sets of dominant integral weights and unordered sets of nonnegative integers, satisfying certain explicit conditions. We describe closures of the strata in terms of representation theory.
• ### The $(\mathfrak{gl}_m,\mathfrak{gl}_n$) duality in the quantum toroidal setting(1801.08433)

Jan. 25, 2018 math.QA
On a Fock space constructed from $mn$ free bosons and lattice ${\Bbb {Z}}^{mn}$, we give a level $n$ action of the quantum toroidal algebra $\mathscr {E}_m$ associated to $\mathfrak{gl}_m$, together with a level $m$ action of the quantum toroidal algebra ${\mathscr E}_n$ associated to ${\mathfrak {gl}}_n$. We prove that the $\mathscr {E}_m$ transfer matrices commute with the $\mathscr {E}_n$ transfer matrices after an appropriate identification of parameters.
• ### Fermionic formulas for eigenfunctions of the difference Toda Hamiltonian(0812.2306)

Jan. 24, 2018 math-ph, math.MP, math.QA
We use the Whittaker vectors and the Drinfeld Casimir element to show that eigenfunctions of the difference Toda Hamiltonian can be expressed via fermionic formulas. Motivated by the combinatorics of the fermionic formulas we use the representation theory of the quantum groups to prove a number of identities for the coefficients of the eigenfunctions.
• ### Branching rules for quantum toroidal gl(n)(1309.2147)

Jan. 13, 2018 math-ph, math.MP, math.QA
We construct an analog of the subalgebra $Ugl(n)\otimes Ugl(m)$ of $Ugl(m+n)$ in the setting of quantum toroidal algebras and study the restrictions of various representations to this subalgebra.
• ### Integrals of motion from quantum toroidal algebras(1705.07984)

Aug. 18, 2017 math-ph, math.MP, math.QA
We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors. That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model. We also discuss the (gl(m),gl(n)) duality of XXZ models in quantum toroidal setting and the implications for the quantum KdV model. In particular, we conjecture that the spectrum of non-local integrals of motion of Bazhanov, Lukyanov, and Zamolodchikov is described by Gaudin Bethe ansatz equations associated to affine sl(2).
• ### On the Gaudin model associated to Lie algebras of classical types(1512.08524)

Oct. 20, 2016 math-ph, math.MP, math.QA
We derive explicit formulas for solutions of the Bethe Ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple.
• ### Finite type modules and Bethe ansatz equations(1609.05724)

Sept. 19, 2016 math-ph, math.MP, math.QA, math.RT
We introduce and study a category $\text{Fin}$ of modules of the Borel subalgebra of a quantum affine algebra $U_q\mathfrak{g}$, where the commutative algebra of Drinfeld generators $h_{i,r}$, corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional $U_q\mathfrak{g}$ modules. In particular, we classify the irreducible objects, discuss their properties, and describe the combinatorics of the q-characters. We study transfer matrices corresponding to modules in $\text{Fin}$. Among them we find the Baxter $Q_i$ operators and $T_i$ operators satisfying relations of the form $T_iQ_i=\prod_j Q_j+ \prod_k Q_k$. We show that these operators are polynomials of the spectral parameter after a suitable normalization. This allows us to prove the Bethe ansatz equations for the zeroes of the eigenvalues of the $Q_i$ operators acting in an arbitrary finite-dimensional representation of $U_q\mathfrak{g}$.
• ### Finite type modules and Bethe Ansatz for quantum toroidal gl(1)(1603.02765)

March 9, 2016 math-ph, math.MP, math.QA
We study highest weight representations of the Borel subalgebra of the quantum toroidal gl(1) algebra with finite-dimensional weight spaces. In particular, we develop the q-character theory for such modules. We introduce and study the subcategory of `finite type' modules. By definition, a module over the Borel subalgebra is finite type if the Cartan like current \psi^+(z) has a finite number of eigenvalues, even though the module itself can be infinite dimensional. We use our results to diagonalize the transfer matrix T_{V,W}(u;p) analogous to those of the six vertex model. In our setting T_{V,W}(u;p) acts in a tensor product W of Fock spaces and V is a highest weight module over the Borel subalgebra of quantum toroidal gl(1) with finite-dimensional weight spaces. Namely we show that for a special choice of finite type modules $V$ the corresponding transfer matrices, Q(u;p) and T(u;p), are polynomials in u and satisfy a two-term TQ relation. We use this relation to prove the Bethe Ansatz equation for the zeroes of the eigenvalues of Q(u;p). Then we show that the eigenvalues of T_{V,W}(u;p) are given by an appropriate substitution of eigenvalues of Q(u;p) into the q-character of V.
• ### Quantum toroidal gl(1) and Bethe ansatz(1502.07194)

Feb. 25, 2015 math-ph, math.MP, math.QA
We establish the method of Bethe ansatz for the XXZ type model obtained from the R-matrix associated to quantum toroidal gl(1). We do that by using shuffle realizations of the modules and by showing that the Hamiltonian of the model is obtained from a simple multiplication operator by taking an appropriate quotient. We expect this approach to be applicable to a wide variety of models.
• ### Lower bounds for numbers of real solutions in problems of Schubert calculus(1404.7194)

We give lower bounds for the numbers of real solutions in problems appearing in Schubert calculus in the Grassmannian Gr(n,d) related to osculating flags. It is known that such solutions are related to Bethe vectors in the Gaudin model associated to gl(n). The Gaudin Hamiltonians are selfadjoint with respect to a nondegenerate indefinite Hermitian form. Our bound comes from the computation of the signature of that form.
• ### Spaces of quasi-exponentials and representations of the Yangian Y(gl_N)(1303.1578)

March 16, 2013 math.AG, math.QA, math.RT
We consider a tensor product $V(b)= \otimes_{i=1}^n\C^N(b_i)$ of the Yangian $Y(gl_N)$ evaluation vector representations. We consider the action of the commutative Bethe subalgebra $B^q \subset Y(gl_N)$ on a $gl_N$-weight subspace $V(b)_\lambda \subset V(b)$ of weight $\lambda$. Here the Bethe algebra depends on the parameters $q=(q_1,...,q_N)$. We identify the $B^q$-module $V(b)_\lambda$ with the regular representation of the algebra of functions on a fiber of a suitable discrete Wronski map. If $q=(1,...,1)$, we study the action of $B^{q=1}$ on a space $V(b)^{sing}_\lambda$ of singular vectors of a certain weight. Again, we identify the $B^{q=1}$-module $V(b)^{sing}_\lambda$ with the regular representation of the algebra of functions on a fiber of another suitable discrete Wronski map. These results we announced earlier in relation with a description of the quantum equivariant cohomology of the cotangent bundle of a partial flag variety and a description of commutative subalgebras of the group algebra of a symmetric group.
• ### Bethe Algebra of Homogeneous XXX Heisenberg Model Has Simple Spectrum(0706.0688)

May 25, 2012 math.CA, math.AG, math.QA
We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional $gl_2$-modules. As a byproduct we show that there exist exactly $\binom {n}{l}-\binom{n}{l-1}$ two-dimensional vector subspaces $V \subset \C[u]$ with a basis $f,g\in V$ such that $\deg f = l, \deg g = n-l+1$ and $f(u)g(u-1) - f(u-1)g(u) = (u+1)^n$.
• ### Representations of quantum toroidal $gl_n$(1204.5378)

April 24, 2012 math.QA
We define and study representations of quantum toroidal $gl_n$ with natural bases labeled by plane partitions with various conditions. As an application, we give an explicit description of a family of highest weight representations of quantum affine $gl_n$ with generic level.
• ### Affinization of category O for quantum groups(1204.2769)

April 12, 2012 math.QA
Let g be a simple Lie algebra. We consider the category O-hat of those modules over the affine quantum group Uq(g-hat) whose Uq(g)-weights have finite multiplicity and lie in a finite union of cones generated by negative roots. We show that many properties of the category of the finite-dimensional representations naturally extend to the category O-hat. In particular, we develop the theory of q-characters and define the minimal affinizations of parabolic Verma modules. In types ABCFG we classify these minimal affinizations and conjecture a Weyl denominator type formula for their characters.
• ### On Conjectures of A. Eremenko and A. Gabrielov(1201.3589)

Jan. 17, 2012 math-ph, math.MP, math.CA, math.QA
We study polynomials p(x) satisfying a differential equation of the form p"(x)-h'(x)p'(x)+H(x)p(x)=0, where h(x)=x^3/3+ax. We prove a conjecture of A. Eremenko and A. Gabrielov.
• ### Extended T-systems(1104.3094)

Jan. 14, 2012 hep-th, math-ph, math.MP, math.QA
We use the theory of q-characters to establish a number of short exact sequences in the category of finite-dimensional representations of the quantum affine groups of types A and B. That allows us to introduce a set of 3-term recurrence relations which contains the celebrated T-system as a special case.
• ### Quantum toroidal $\mathfrak{gl}_1$ algebra : plane partitions(1110.5310)

Oct. 24, 2011 math-ph, math.MP, math.QA
In third paper of the series we construct a large family of representations of the quantum toroidal $\gl_1$ algebra whose bases are parameterized by plane partitions with various boundary conditions and restrictions. We study the corresponding formal characters. As an application we obtain a Gelfand-Zetlin type basis for a class of irreducible lowest weight $\gl_\infty$-modules.
• ### Path description of type B q-characters(1103.5873)

March 30, 2011 math.CO, math.QA
We give a set of sufficient conditions for a Laurent polynomial to be the q-character of a finite-dimensional irreducible representation of a quantum affine group. We use this result to obtain an explicit path description of q-characters for a class of modules in type B. In particular, this proves a conjecture of Kuniba-Ohta-Suzuki.
• ### Reality property of discrete Wronski map with imaginary step(1102.5368)

Feb. 25, 2011 math-ph, math.MP, math.AG, math.QA
For a set of quasi-exponentials with real exponents, we consider the discrete Wronskian (also known as Casorati determinant) with pure imaginary step 2h. We prove that if the coefficients of the discrete Wronskian are real and for every its roots the imaginary part is at most |h|, then the complex span of this set of quasi-exponentials has a basis consisting of quasi-exponentials with real coefficients. This result is a generalization of the statement of the B. and M. Shapiro conjecture on spaces of polynomials. The proof is based on the Bethe ansatz for the XXX model.
• ### Quantum continuous $\mathfrak{gl}_\infty$: Semi-infinite construction of representations(1002.3100)

Feb. 16, 2010 math.QA, math.RT
We begin a study of the representation theory of quantum continuous $\mathfrak{gl}_\infty$, which we denote by $\mathcal E$. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference operators acting on the space of Laurent polynomials in one variable. Fundamental representations of $\mathcal E$ are labeled by a continuous parameter $u\in {\mathbb C}$. The representation theory of $\mathcal E$ has many properties familiar from the representation theory of $\mathfrak{gl}_\infty$: vector representations, Fock modules, semi-infinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from $\mathcal E$ to spherical double affine Hecke algebras $S\ddot H_N$ for all $N$. A key step in this construction is an identification of a natural bases of the tensor products of vector representations with Macdonald polynomials. We also show that one of the Fock representations is isomorphic to the module constructed earlier by means of the $K$-theory of Hilbert schemes.
• ### Quantum continuous $gl_\infty$: Tensor products of Fock modules and $W_n$ characters(1002.3113)

Feb. 16, 2010 hep-th, math-ph, math.MP, math.QA, math.RT
We construct a family of irreducible representations of the quantum continuous $gl_\infty$ whose characters coincide with the characters of representations in the minimal models of the $W_n$ algebras of $gl_n$ type. In particular, we obtain a simple combinatorial model for all representations of the $W_n$-algebras appearing in the minimal models in terms of $n$ interrelating partitions.
• ### Bethe algebra of the gl_{N+1} Gaudin model and algebra of functions on the critical set of the master function(0910.4690)

Oct. 24, 2009 math-ph, math.MP, math.AG, math.QA
Consider a tensor product of finite-dimensional irreducible gl_{N+1}-modules and its decomposition into irreducible modules. The gl_{N+1} Gaudin model assigns to each multiplicity space of that decomposition a commutative (Bethe) algebra of linear operators acting on the multiplicity space. The Bethe ansatz method is a method to find eigenvectors and eigenvalues of the Bethe algebra. One starts with a critical point of a suitable (master) function and constructs an eigenvector of the Bethe algebra. In this paper we consider the algebra of functions on the critical set of the associated master function and show that the action of this algebra on itself is isomorphic to the action of the Bethe algebra on a suitable subspace of the multiplicity space. As a byproduct we prove that the Bethe vectors corresponding to different critical points of the master function are linearly independent and, in particular, nonzero.
• ### Three sides of the geometric Langlands correspondence for gl_N Gaudin model and Bethe vector averaging maps(0907.3266)

July 19, 2009 math-ph, math.MP, math.QA
We consider the gl_N Gaudin model of a tensor power of the standard vector representation. The geometric Langlands correspondence in the Gaudin model relates the Bethe algebra of the commuting Gaudin Hamiltonians and the algebra of functions on a suitable space of N-th order differential operators. In this paper we introduce a third side of the correspondence: the algebra of functions on the critical set of a master function. We construct isomorphisms of the third algebra and the first two. A new object is the Bethe vector averaging maps.


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## Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem.(English)Zbl 1032.34040

The author provides new existence results for the periodic boundary value problem $x''=f(t,x), \quad x(0)=x(T),\;x'(0)=x'(T), (1)$ where $$f$$ is a Carathéodory function. The proofs are based on the Krasnoselskii fixed-point theorem for completely continuous operators in a Banach space that exhibits a cone compression and expansion, and on the sign behaviour of Green’s function of the linearized equation.
The main results are contained in two theorems which give conditions guaranteeing the existence of a positive solution to (1). Modified assertions for negative solutions are shown.
As applications of these general results, the author obtains new existence results for equations with jumping nonlinearities and for equations with a repulsive or attractive singularity in the origin. Weak singularities are considered here, too.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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### References:

 [1] Bevc, V.; Palmer, J.L.; Süsskind, C., On the design of the transition region of axi-symmetric magnetically focusing beam valves, J. br inst. radio eng., 18, 696-708, (1958) [2] Ding, T., A boundary value problem for the periodic Brillouin focusing system, Acta sci. natur. univ. pekinensis, 11, 31-38, (1965), (in Chinese) [3] De Coster, C.; Habets, P., Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results, (), 1-78 · Zbl 0889.34018 [4] del Pino, I.; Manásevich, R.; Montero, A., T-periodic solutions for some second order differential equations with singularities, Proc. roy. soc. edinbourgh sect. A, 120, 231-243, (1992) · Zbl 0761.34031 [5] Drábek, P., Landesman-lazer condition for nonlinear problems with jumping nonlinearities, J. differential equations, 85, 186-199, (1990) · Zbl 0699.34019 [6] Esmailzadeh, E.; Nakhaie-Jazar, G., Periodic solutions of a Mathieu-Duffing type equation, J. non. linear mech., 32, 905-912, (1997) · Zbl 0881.34064 [7] Erbe, L.H.; Mathsen, R.M., Positive solutions for singular nonlinear boundary value problems, Nonlinear anal., 46, 979-986, (2001) · Zbl 1007.34020 [8] Erbe, L.H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proc. amer. math. soc., 120, 743-748, (1994) · Zbl 0802.34018 [9] Fonda, A., Periodic solutions of scalar second order differential equations with a singularity, Mém. classe sci. acad. roy. belgique, 8-IV, 68-98, (1993) [10] Fonda, A.; Habets, P., Periodic solutions of asymptotically positively homogeneous differential equations, J. differential equations, 81, 68-98, (1989) · Zbl 0692.34041 [11] Friedman, B., Principles and techniques of applied mathematics, (1990), Dover Publications Inc New York · Zbl 1227.00033 [12] Gaete, S.; Manasevich, R.F., Existence of a pair of periodic solutions of an O.D.E. generalizing a problem in nonlinear elasticity, via variational methods, J. math. anal. appl., 134, 257-271, (1988) · Zbl 0672.34030 [13] Guo, D.; Laksmikantham, V., Multiple solutions of two-point boundary value problems of ordinary differential equations in Banach spaces, J. math. anal. appl., 129, 211-222, (1988) · Zbl 0645.34014 [14] Habets, P.; Sanchez, L., Periodic solution of some Liénard equations with singularities, Proc. amer. math. soc., 109, 1135-1144, (1990) [15] Henderson, J.; Wang, H., Positive solutions for nonlinear eigenvalue problems, J. math. anal. appl., 208, 252-259, (1997) · Zbl 0876.34023 [16] Ianacci, R.; Nkashama, M.N.; Omari, P.; Zanolin, F., Periodic solutions of forced Liénard equations with jumping nonlinearities under nonuniform conditions, Proc. roy. soc. Edinburgh sect. A, 110, 183-198, (1988) · Zbl 0693.34046 [17] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen · Zbl 0121.10604 [18] Lazer, A.C.; McKenna, P.J., Large-amplitude periodic oscillations in suspension bridgessome new connections with nonlinear analysis, SIAM rev., 32, 537-578, (1990) · Zbl 0725.73057 [19] Lazer, A.C.; Solimini, S., On periodic solutions of nonlinear differential equations with singularities, Proc. amer. math. soc., 99, 109-114, (1987) · Zbl 0616.34033 [20] R. Martins, Existence of periodic solutions for second-order differential equations with singularities and the strong force condition, preprint. · Zbl 1100.34033 [21] Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, (), 74-142 · Zbl 0798.34025 [22] Merivenci Atici, F.; Guseinov, G.Sh., On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions, J. comput. appl. math., 132, 341-356, (2001) · Zbl 0993.34022 [23] Omari, P.; Ye, W.-Y., Necessary and sufficient conditions for the existence of periodic solutions of second-order ordinary differential equations with singular nonlinearities, Differential integral equations, 8, 1843-1858, (1995) · Zbl 0831.34048 [24] Rachunková, I., Existence of two positive solutions of a singular nonlinear periodic boundary value problems, J. differential equations, 176, 445-469, (2001) · Zbl 1004.34008 [25] Rachunková, I.; Tvrdý, M.; Vrkoc̆, I., Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. differential equations, 176, 445-469, (2001) · Zbl 1004.34008 [26] Torres, P., Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system, Math. methods appl. sci., 23, 1139-1143, (2000) · Zbl 0966.34038 [27] P. Torres, M. Zhang, A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. Nach., to appear. · Zbl 1024.34030 [28] Ye, Y.; Wang, X., Nonlinear differential equations in electron beam focusing theory, Acta math. appl. sinica, 1, 13-41, (1978), (in Chinese) [29] Yujun, D., Invariance of homotopy and an extention of a theorem by habets-metzen on periodic solutions of Duffing equations, Nonlinear anal., 46, 1123-1132, (2001) · Zbl 1005.34011 [30] Zhang, M., Periodic solutions of Liénard equations with singular forces of repulsive type, J. math. anal. appl., 203, 254-269, (1996) · Zbl 0863.34039 [31] Zhang, M., A relationship between the periodic and the Dirichlet BVPs of singular differential equations, Proc. roy. soc. edinbourgh sect. A, 128, 1099-1114, (1998) · Zbl 0918.34025 [32] Zhang, M.; Li, W.G., A Lyapunov-type stability criterion using Lα norms, Proc. amer. math. soc., 130, 11, 3325-3333, (2002) · Zbl 1007.34053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

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## Tuesday, September 01, 2009

### Quotient Rings

The following definitions and lemmas are taken from Jean Tignol's Galois' Theory of Algebraic Equations.

In a previous blogs, I wrote about cosets and ideals. In today's blog, I will show how we can bring these ideas together to define quotient rings. Here are links to review the properties of groups, subgroups, or commutative rings.

Definition 1: A/I

A/I = { a + I such that a ∈ A}

Note: This is a set of sets. For example, if a + I is a coset, then A/I is the set of distinct cosets. For review of the a + I notation, see here.

Example 1.1: Modular sets

The sets Z/nZ are all examples of A/I.

Z/2Z = { 0+2z, 1+2z } since { 0 + 2z = 2 + 2z = 2z + 2z, 1 + 2z = 3 + 2z = ... }

Z/3Z = { 0 + 3z, 1 + 3z, 2 + 3z }

A/I becomes especially interesting when A is a Commutative Ring and I is in an Ideal. I will assume both of these properties for the rest of this article.

(a + I) + (b + I) = (a + b) + I

Definition 3: Multiplication for A/I

(a + I) * (b + I) = ab + I

Lemma 1: Addition for A/I is well defined

Proof:

(1) Let:

s + I = s' + I

t + I = t' + I

(2) Using Lemma 1, here, it follows that:

s -s' ∈ I

and

t - t' ∈ I

(3) So, there exists a,b such that a, b ∈ I and:

s = s' + a

t = t' + b

(4) s + t = (s' + a) + (t' + b) = s' + a + t' + b

(5) s + t + I = s' + t' + (a + b) + I

(6) Since a ∈ I and b ∈ I, it follows that a + b ∈ I (from Closure)

(7) Using Lemma 2, here, we then have:

(a+b) + I = I

(8) So that:

s + t + I = s' + t' + I

QED

Lemma 2: Multiplication for A/I is well defined

Proof:

(1) Assume that I is an ideal.

(2) Let:

s + I = s' + I t + I = t' + I

(3) Using Lemma 1, here, it follows that:

s -s' ∈ I

and

t - t' ∈ I

(4) There exists a,b such that a, b ∈ I and:

s = s' + a t = t' + b

(5) st = (s' + a)(t' + b) = s't' + at' + s'b + ab

(6) st + I = s't' + at' + s'b + ab + I

(7) Since a,t',s',b ∈ I, we have (see Definition 2, here, and Definition 1, here for details):

at' ∈ I

s'b ∈ I

ab ∈ I

(8) So, at' + s'b + ab + I = I [See Lemma 2, here]

(9) And:

st + I = s't' + I

QED

Lemma 3: If A is a Commutative Ring and I is an Ideal, then A/I is a ring

Proof:

(a + I) + (b + I) = (a + b) + I = (b + a) + I = (b + I) + (a + I)

[(a + I) + (b + I)] + (c + I) = (a + b) + I + c + I = (a + b + c) + I = a + I + (b + c) + I = (a + I) + [(b + I) + (c + I)]

0+I is the additive identity since 0 ∈ A and for all a, (a + I) + (0 + I) = (a + 0) + I = a + I

-a+I is the additive inverse since a ∈ A → -a ∈ A and (a + I) + (-a + I) = (a + -a) + I = 0 + I

(5) Associative Rule for Multiplication

[(a + I)*(b+I)](c + I) = (ab + I)(c + I) = (abc + I) = (a + I)(bc + I) = (a+I)[(b+I)(c+I)]

(6) Distributive Rule

(a + I)[(b + I) + (c + I) ] = [(a + I)(b+I)] + [(a+I)(c+I)] = (ab + I) + (ac + I)

QED

Definition 5: Quotient Ring

The ring A/I is called the quotient ring of A by the ideal I.

Example 5.1: Sets that form quotient rings

Z/2Z and Z/3Z are factor rings [See Example 1.1 above for details]

Reference
• Jean-Pierre Tignol, , World Scientific, 2001

## Monday, August 31, 2009

### Cosets

The following definitions and lemmas are taken from Jean Tignol's Galois' Theory of Algebraic Equations.

Definition 1: g + H

Let g + H = { g + h such that h ∈ H }.

Example 1.1:

Let G be the set { 0, 1, 2, 3 }

Let H be the set { 9, 10, 11 }

Then:

0 + H = { 9, 10, 11 }

1 + H = { 10, 11, 12 }

2 + H = { 11, 12, 13 }

3 + H = { 12, 13, 14 }

Definition 2: coset of H in G

The set g + H is a coset if and only if there exists a group G such that g ∈ G and H is a subgroup of G.

Note: Technically, definition 3 above describes a left coset. A coset can also be defined as H + g which is called a right coset. The notation of g + H defines a coset based on addition. Cosets can also be defined on multiplication and represented as gH or Hg.

Example 2.1:

Let G = the set of integers Z
Let H = 2Z = the set of even integers

1 ∈ Z
1+H = odd integers = { ... -3, -1, 1, 3, ... }

Example 2.2: Cyclic Group Z4

Z4 = { 0, 1, 2, 3}

It is a group since:

(1) Closure

Since Z4 is always modulo 4, it is clear that that operation of addition is closed (for example: 3 + 2 = 1)

(2) Associativity

For any elements a,b,c ∈ Z4, (a + b) + c = a + (b + c)

(3) Identity Element

0 is the identity element

(4) Inverse Element

Since it is modulo 4, each element has an inverse: 0+0=0, 1+3=0, 2+2=0

Let H = { 0 , 2 }

H is a subset and H is itself a group since:

(1) It has closure: 0+2=2, 0+0=0, 2+2=0

(2) It has associativity.

(3) It has a identity element: 0

(4) From #1, it is clear that each element is its own inverse.

From Z4 and H, there are 2 distinct cosets:

0 + H = { 0, 2}
1 + H = { 1, 3}
2 + H = { 2, 0} = 0 + H
3 + H = { 3, 1} = 1 + H

Lemma 1:

For the coset H in G:

For all a,b ∈ G:

a + H = b + H if and only if a - b ∈ H

Proof:

(1) Assume that a + H = b + H

(2) So, for all x ∈ (a + H) → x ∈ (b + H)

(3) a ∈ a + H, b ∈ b + H [since H is a group and therefore 0 ∈ H]

(4) a ∈ b + H [follows directly from step #2]

(5) Let c = a - b

(6) b + c ∈ b + H [since b + c = a and a ∈ b + H]

(7) So c ∈ H [from step #6 and Definition 3 above]

(8) Assume that (a-b) ∈ H

(9) Assume that x ∈ a + H

(10) Let y = x - a

(11) y ∈ H [since x = a + y]

(12) y + (a - b) ∈ H since H is a group

(13) x - b ∈ H [since y + (a - b) = (x - a) + (a - b) = x - b]

(14) Then x ∈ b + H since [b + x-b = x]

(15) We can make the same argument if x ∈ b + H so this shows that a + H = b + H.

QED

Lemma 2:

For the coset H in G:

a ∈ H → a + H = H

Proof:

(1) Assume that a ∈ H

(2) Assume that x ∈ a + H

(3) Then there exists h such that x = a + h where h ∈ H [See Definition 1 above]

(4) But if a ∈ H and h ∈ H, then a + h ∈ H. [since H is a group, see Definition 2 above and the Closure property of groups]

(5) So x ∈ H

(6) Assume that x ∈ H

(7) Let b = x - a

(8) Since a ∈ H, it follows that -a ∈ H [See Definition 2 above and the Inverse property of groups]

(9) Since x ∈ H and -a ∈ H, it follows that b ∈ H [from the Closure property of groups]

(10) So then, x ∈ a + H [since b ∈ H and x = a + b from Definition 1 above]

QED

Reference
• Jean-Pierre Tignol, , World Scientific, 2001

## Sunday, August 30, 2009

### Ideals

The following definitions and lemmas are taken from Jean Tignol's Galois' Theory of Algebraic Equations.

Definition 1: stable under multiplication

a set I is stable under multiplication by the elements in A if and only if:

a ∈ A and x ∈ I ↔ ax ∈ I.

Example 1.1: set that is stable under multiplication

A = {1}

I = {1}

Example 1.2: set that is not stable under multiplication

A= {1,2}

I = {2}

2 ∈ A and 2 ∈ I but 2*2=4 is not in I.

Definition 2: An ideal I

Let A be a commutative ring.

a set I is an ideal if and only if I is a subgroup of the additive group of A which is stable under multiplication by the elements in A (see Definition 1 above).

Example 2.1: Example of a set that is an ideal: 2Z

The set 2Z is the set of even integers.

2Z = { ... -2, 0, 2, 4, 6, ... }

2Z is a subset of the set of integers Z and Z is a commutative ring and a group.

2Z is a group on the operation of addition so it is a subgroup of Z.

2Z is stable under multiplication by elements of Z since an even integer multiplied by any integer is an even integer.

Example 2.2: Example of a set that is not an ideal: 2W

Let 2W be the set of even whole numbers = { 0, 2, 4, ... }

2W is a subset of the set of integers Z which is a commutative ring and a group.

2W is not a group on the operation of addition since there is no inverse element.

2W is not stable under multiplication by the elements of Z since -1 ∈ Z and 2 ∈ 2W but -2 is not in 2W.

Lemma 1:

The set of multiples for a given polynomial is an ideal of the set of all polynomials for a given field

Proof:

(1) Let F[X] be the set of all polynomials in the field F

(2) Let (P) be the set of multiples of polynomials for a given polynomial P so that:

(P) = { PQ where Q ∈ F[X] }

(3) All fields are commutative rings so F[X] is a Commutative Ring. [See Definition 3, here for Fields]

(4) (P) is itself a group since:

If pq, pq' ∈ (P), then (pq+pq')=p(q+q') ∈ (P) since (q+q') ∈ F[X]

(pq + pq') + pq'' = p([q + q'] + q'') = p(q + [q' + q'']) = pq + (pq' + pq'')

(c) Identity Element

0 = P*0 ∈ (P) since 0 ∈ F[X]

(d) Inverse Element

For all pq, there exists -pq since -pq = p(-q) and -q ∈ F[X] if q ∈ F[X]

(5) Finally, (P) is stable under multiplication since:

(a) Assume pq, pq' ∈ (P)

(b) Then q,q' ∈ F[X]

(c) p ∈ F[X]

(d) So pqq' ∈ F[X] since F[X] is closed on multiplication.

(e) So p*(pqq') ∈ (P)

QED

Reference
• Jean-Pierre Tignol, , World Scientific, 2001


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# Basis of coinvariant algebra on which reflection group acts as regular representation

This question is almost a duplicate of a question of Christian Stump, except that Christian seems to ask about an isomorphism to irreducible representations rather than the regular representation: What does the regular representation of the coinvariant ring of a unitary reflection group look like?

If $W\subseteq GL(V)$ is a finite reflection group, then the Chevalley-Shephard-Todd theorem says that the invariant algebra $\mathbb{C}[V]^W$ is isomorphic to a polynomial algebra $\mathbb{C}[V]^W\simeq \mathbb{C}[e_1,e_2,\ldots,e_n]$ and that the coinvariant algebra $\mathbb{C}[V]^{\mathrm{co}W} := \mathbb{C}[V]/\mathbb{C}[V]^W_+$ (where $\mathbb{C}[V]^W_+$ is the set of invariant polynomials of positive degree) is isomorphic as a $W$-module to the left regular representation.

Question: can we write down some explicit basis of $\mathbb{C}[V]^{\mathrm{co}W}$ on which $W$ acts as the regular representation?

The only proofs I know of the fact that $\mathbb{C}[V]^{\mathrm{co}W}$ carries the regular representation use character computations which have a "non-constructive" flavor.

Really I am most interested just in the case of the symmetric group $W=S_n$.

Let me give a quick example of what this looks like for $W=S_2$. Then $\mathbb{C}[V] = \mathbb{C}[x_1,x_2]$ and we get $\mathbb{C}[V]^W = \mathbb{C}[x_1+x_2,x_1x_2]$ (these are the elementary symmetric polynomials''). (Maybe strictly speaking because $S_2$ acts on $\mathbb{R}^2/(1,1)$ I should write $\mathbb{C}[V] = \mathbb{C}[x_1,x_2]/\langle x_1+x_2 \rangle$ and $\mathbb{C}[V]^W =\mathbb{C}[x_1x_2]$ but I don't think this technicality matters.) At any rate we have that the coinvariant ring is $\mathbb{C}[V]^{\mathrm{co}W}=\mathbb{C}[x_1,x_2]/\langle x_1+x_2,x_1x_2 \rangle$. There are standard bases of the coinvariant ring for the symmetric group, like the staircase monomials or the Schubert polynomials. In this case both of those bases would be $\{x_1,1\}$ (note that those bases are homogeneous). But the symmetric group $S_2$ does not act on that basis as in the regular representation. Instead I would want a basis like $\{x_1+1,-x_1+1\}$.

EDIT: Christian Gaetz asked me what taking $W=C_n=\langle c \rangle$ the cyclic group of order $n$ looks like. Here $c$ acts on $\mathbb{C}$ by $c\cdot x = \xi x$ with $\xi$ a primitive $n$th root of unity. Thus $\mathbb{C}[V]^{\mathrm{co}W} = \mathbb{C}[x]/\langle x^n \rangle$. A good choice of basis here is $\{c^m\cdot f\colon m=0,1,\ldots,n-1\}$ where $f=1+x+x^2+\cdots+x^{n-1}$. Indeed, in this case you can check that all the $c^m\cdot f$ are linearly independent by writing them in the standard basis $\{1,x,\ldots,x^{n-1}\}$ and then evaluating a Vandermonde determinant where you plug in $x_i=\xi^i$.

• Even for $W = S_n$, this question is extremely interesting! The descent basis seems to be a step in the right direction, but still not what we're looking for. – darij grinberg Apr 30 '18 at 18:10
• @darijgrinberg: the descent basis still has the problem that it is homogenous, correct? – Sam Hopkins May 1 '18 at 13:56
• True... I guess this is why those bases are so hard to find. – darij grinberg May 1 '18 at 15:03
• I think that maybe deforming the harmonic polynomials / space of all iterated partial derivatives of the Vandermonde in the case of $S_n$ is promising. At least it is easy to see how to try and make things inhomogeneous, by doing things like $\partial_i+a$ instead of $\partial_i$. – Vladimir Dotsenko May 1 '18 at 16:35
• Another remark: while indeed many/most proofs compute characters, for $S_n$ this can alternatively be proved using Galois theory. Namely, take $L=\mathbb{C}(x_1,\ldots,x_n)$. The field of $S_n$-invariants of this would be rational symmetric functions which are $K=\mathbb{C}(e_1,\ldots,e_n)$. We see that $[L:K]=n!$ and in fact is the regular representation by the normal basis theorem. Passing from that result to the statement about the rings before taking fields of fractions is an easy exercise :-) – Vladimir Dotsenko May 1 '18 at 16:42

Any basis $B$ on which $W$ acts as the regular representation has the form $\{w\cdot f\,\mid\, w\in W\}$ for some $f$ in the coinvariant ring. Hence it suffices to take $f$ generic. It would be interesting to find some "nice" nongeneric $f$.

• I agree with this but was hoping there might be a more canonical choice. – Sam Hopkins Apr 30 '18 at 18:55

Let $W$ be a finite reflection group. Set $R={\mathbb C}[V]$, $R^W\subset R$ the invariant subring, and $R_W={\mathbb C}[V]^{coW}$ the coinvariant ring. For a linear character $\chi\colon W\rightarrow {\mathbb C}^*$, define the relative invariant ring $R^W_\chi=\left\{g\in R\left|w\cdot g=\chi(w)\cdot g, \ \forall w\in W\right.\right\}$. For reflection groups, this is always a cyclic module over $R^W$ (e.g. see Professor Stanley's paper). If $\chi_1,\ldots,\chi_m$ are the distinct linear characters of $W$, let $f_1,\ldots,f_m$ be any generators of the respective relative invariant rings. For $f\in R$, let $\bar{f}\in R_W$ denote its equivalence class.

Claim: If $W$ is abelian, then the elements $\left\{\sum_{i=1}^m\chi_i(w)\cdot \bar{f}_i\left|w\in W\right.\right\}$ form a basis for $R_W$ on which $W$ acts by the regular representation.

Proof: Note that the equivalence classes $\left\{\bar{f}_1,\ldots,\bar{f}_m\right\}$ are linearly independent because each is a simultaneous eigenvector for $W$ corresponding to a distinct linear character. Since $W$ is abelian this must be a basis for $R_W$. To see that the elements $\left\{\left.\sum_{i=1}^m\chi_i(w)\bar{f}_i\right|w\in W\right\}$ are linearly independent in $R_W$, use linear independence of distinct linear characters.

For non-abelian groups the elements $\left\{\bar{f}_1,\ldots,\bar{f}_m\right\}$ won't be a basis (since $m<|W|$), which will force the set $\left\{\sum_{i=1}^m\chi_i(w)\bar{f}_i\left|w\in W\right.\right\}$ to be linearly dependent.

• The definition of $R_\chi^W$ is not correct. – Richard Stanley May 1 '18 at 2:03
• @RichardStanley thanks, I think the definition is correct now. But the conjecture is still not true in general, although maybe it's true for Abelian groups... – Chris McDaniel May 1 '18 at 2:33
• @SamHopkins I edited my answer. – Chris McDaniel May 2 '18 at 3:58


text:
# Vignette for survRM2 package: In survRM2: Comparing Restricted Mean Survival Time

## 1 Introduction

In a comparative, longitudinal clinical study, often the primary endpoint is the time to a specific clinical event, such as death, heart failure hospitalization, tumor progression, and so on. The hazard ratio estimate is almost routinely used to quantify the treatment difference. However, the clinical meaning of such a model-based between-group summary can be rather difficult to interpret when the underlying model assumption (i.e., the proportional hazards assumption) is violated, and it is difficult to assure that the modeling is indeed correct empirically. For example, a non-significant result of a goodness-of-fit test does not necessary mean that the proportional hazards assumption is "correct." Other issues on the hazard ratio is seen elsewhere [1, 2]. Between-group summery metrics based on the restricted mean survival time (RMST) are useful alternatives to the hazard ratio or other model-based measures. This vignette is a supplemental documentation for survRM2 package and illustrates how to use the functions in the package to compare two groups with respect to the restricted mean survival time. The package was made and tested on R version 3.3.2.

## 2 Sample data

Throughout this vignette, we use a part of data from the primary biliary cirrhosis (pbc) study conducted by the Mayo Clinic, which is included in survival package in R. The details of the study and the data elements are seen in the help file in survival package, which can be seen by

library(survival)
?pbc

The original data in the survival package consists of data from 418 patients, which includes those who participated in the randomized clinical trial and those who did not. In the following illustration, we use only 312 cases who participated in the randomized trial (158 cases on D-penicillamine group and 154 cases on Placebo group). The following function in survRM2 package creates the data used in this vignette, selecting the subset from the original data file.

library(survRM2)

D = rmst2.sample.data()
nrow(D)

Here, time is years from the registration to death or last known alive, status is the indicator of the event (1: death, 0: censor), and arm is the treatment assignment indicator (1: D-penicillamine, 0: Placebo). Below is the Kaplan-Meier (KM) estimate for time-to-death of each treatment group.

plot(survfit(Surv(time, status)~arm, data=D), col=c("blue","red"), lwd=2, mark.time=F, xlab="Years",ylab="Probability")
legend("bottomleft", c("Placebo (arm=0)","D-penicillamine (arm=1)"), col=c("blue","red"), lwd=2)

## 3 Restricted mean survival time (RMST) and restricted mean time lost (RMTL)

The RMST is defined as the area under the curve of the survival function up to a time $\tau (< \infty):$ $$\mu_{\tau} = \int_0^{\tau} S(t)dt,$$ where $S(t)$ is the survival function of a time-to-event variable of interest. The interpretation of the RMST is that "when we follow up patients for $\tau,$ patients will survive for $\mu_{\tau}$ on average," which is quite straightforward and clinically meaningful summary of the censored survival data. If there were no censored observations, one could use the mean survival time $$\mu_{\infty} = \int_0^{\infty} S(t)dt,$$ instead of $\mu_{\tau}.$

A natural estimator for $\mu_{\tau}$ is $$\hat{\mu}{\tau} = \int_0^{\tau} \hat{S}(t)dt,$$ where $\hat{S}(t)$ is the KM estimator for $S(t).$ The standard error for $\hat{\mu}{\tau}$ is also calculated analytically; the detailed formula is given in [3]. Note that $\mu_{\tau}$ is estimable even under a heavy censoring case. On the other hand, although median survival time, $S^{-1}(0.5),$ is also a robust summary of survival time distribution, it will become inestimable when the KM curve does not reach 0.5 due to heavy censoring or rare events.

The RMTL is defined as the area "above" the curve of the survival function up to a time $\tau:$ $$\tau - \mu_{\tau} = \int_0^{\tau} { 1-S(t) }dt.$$ In the following figure, the area highlighted in pink and orange are the RMST and RMTL estimates, respectively, in D-penicillamine group, when $\tau$ is 10 years. The result shows that the average survival time during 10 years of follow-up is 7.15 years in the D-penicillamine group. In other words, during the 10 years of follow-up, patients treated by D-penicillamine lost 2.85 years in average sense.

fit=survfit(Surv(D$time[D$arm==1], D$status[D$arm==1])~1)
tau=10
tmp.xx=c(0, fit$time); tmp.yy=c(1, fit$surv) ;
idx=tmp.xx<=tau
y.tau = min(tmp.yy[idx])
xx=c(tmp.xx[idx],   tau)
yy=c(tmp.yy[idx], y.tau)
x.step=sort(c(0, tmp.xx, tmp.xx))
y.step=rev(sort(c(1,1,tmp.yy, tmp.yy[-length(tmp.yy)])))

rmst=summary(fit, rmean=10)$table[5]  par(mfrow=c(1,2)) #-------- plot(fit, mark.time=F, xlab="Years",ylab="Probability",conf.int=F, lwd=2, main="Restricted mean survival time (RMST)", col="red", cex.main=0.8) for (i in 1: (length(xx)-1)){ polygon(c(xx[i], xx[i+1], xx[i+1], xx[i]), c(0, 0, yy[i+1], yy[i]), col="pink", density=80, angle=80, lwd=2) } lines(x.step, y.step, col="red", lwd=3) text(5,0.4, paste(round(rmst, digits=2),"years"), cex=0.9) #-------- plot(fit, mark.time=F, xlab="Years",ylab="Probability", conf.int=F, lwd=2, main="Restricted mean time lost (RMTL)", col="red",cex.main=0.8) for (i in 1: (length(xx)-1)){ polygon(c(xx[i], xx[i+1], xx[i+1], xx[i]), c(yy[i], yy[i+1], 1,1), col="orange", density=80, angle=80, lwd=2) } lines(x.step, y.step, col="red", lwd=3) text(7,0.8, paste(round(tau-rmst, digits=2),"years"), cex=0.9)  ### 3.1 Unadjusted analysis and its implementation Let$\mu_{\tau}(1)$and$\mu_{\tau}(0)$denote the RMST for treatment group 1 and 0, respectively. Now, we compare the two survival curves, using the RMST or RMTL. Specifically, we consider the following three measures for the between-group contrast. 1. Difference in RMST $$\mu_{\tau}(1) - \mu_{\tau}(0)$$ 2. Ratio of RMST $$\mu_{\tau}(1) / \mu_{\tau}(0)$$ 3. Ratio of RMTL $${ \tau - \mu_{\tau}(1) } / { \tau - \mu_{\tau}(0) }$$ These are estimated by simply replacing$\mu_{\tau}(1)$and$\mu_{\tau}(0)$by their empirical counterparts (i.e.,$\hat{\mu}{\tau}(1)$and$\hat{\mu}{\tau}(0)$, respectively). For inference of the ratio type metrics, we use the delta method to calculate the standard error. Specifically, we consider$\log { \hat{\mu}{\tau}(1) }$and$\log { \hat{\mu}{\tau}(0) }$and calculate the standard error of log-RMST. We then calculate a confidence interval for log-ratio of RMST, and transform it back to the original ratio scale. Below shows how to use the function, rmst2, to implement these analyses. time = D$time
status = D$status arm = D$arm

rmst2(time, status, arm, tau=10)

The first argument (time) is the time-to-event vector variable. The second argument (status) is also a vector variable with the same length as time, each of the elements takes either 1 (if event) or 0 (if no event). The third argument (arm) is a vector variable to indicate the assigned treatment of each subject; the elements of this vector take either 1 (if active treatment arm) or 0 (if control arm). The fourth argument (tau) is a scalar value to specify the truncation time point ${\bf \tau}$ for the RMST calculation. Note that $\tau$ needs to be smaller than the minimum of the largest observed time in each of the two groups (let us call this the max $\tau$). The program will stop with an error message when such $\tau$ is specified. When $\tau$ is not specified in rmst2, i.e., when the code looks like

rmst2(time, status, arm)

the max $\tau$ is used as the default $\tau.$ It is always encouraged to confirm that the size of the risk set is large enough at the specified $\tau$ in each group to make sure the stability of the KM estimates.

Below is the output with the pbc example when $\tau=10$ (years) is specified. The rmst2 function returns RMST and RMTL on each group and the results of the between-group contrast measures listed above.

obj = rmst2(time, status, arm, tau=10)
print(obj)

In the present case, the difference in RMST (the first row of the "Between-group contrast" block in the output) was -0.137 years. The point estimate indicated that patients on the active treatment survive 0.137 years shorter than those on placebo group on average, when following up the patients 10 years. While no statistical significance was observed (p=0.738), the 0.95 confidence interval (-0.665 to 0.939) was relatively tight around 0, suggesting that the difference in RMST would be at most +/- one year.

The package also has a function to generate a plot from the rmst2 object. The following figure is automatically generated by simply passing the resulting rmst2 object to plot() function after running the aforementioned unadjusted analyses.

plot(obj, xlab="Years", ylab="Probability")

### 3.2 Adjusted analysis and implementation

In most of the randomized clinical trials, an adjusted analysis is usually included in one of the planned analyses. One reason would be that adjusting for important prognostic factors may increase power to detect a between-group difference. Another reason would be we sometimes observe imbalance in distribution of some of baseline prognostic factors even though the randomization guarantees the comparability of the two groups on average. The function, rmst2, in this package implements an ANCOVA type adjusted analysis proposed by Tian et al. [4], in addition to the unadjusted analyses presented in the previous section.

Let $Y$ be the restricted mean survival time, and let $Z$ be the treatment indicator. Also, let $X$ denote a $q$-dimensional baseline covariate vector. Tian's method consider the following regression model, $$g{ E(Y \mid Z, X) } = \alpha + \beta Z + \gamma^\prime X,$$ where $g(\cdot)$ is a given smooth and strictly increasing link function, and $(\alpha, \beta, \gamma^\prime)$ is a $(q+2)$-dimension unknown parameter vector. Prior to Tian et al. [4], Andersen et al. [5] also studied this regression model and proposed an inference procedure for the unknown model parameter, using a pseudo-value technique to handle censored observations. Program codes for their pseudo-value approach are available on the three major platforms (Stata, R and SAS) with detailed documentation [6, 7]. In contrast to Andersen's method [5, 6, 7], Tian's method [4] utilizes an inverse probability censoring weighting technique to handle censored observations. The function, rmst2, in this package implements this method.

As shown below, for implementation of Tian's adjusted analysis for the RMST, the only the difference is if the user passes covariate data to the function. Below is a sample code to perform the adjusted analyses.

rmst2(time, status, arm, tau=10, covariates=x)

where covariates is the argument for a vector/matrix of the baseline characteristic data, x. For illustration, let us try the following three baseline variables, in the pbc data, as the covariates for adjustment.

x=D[,c(4,6,7)]

x=D[,c(4,6,7)]

The rmst2 function fits data to a model for each of the three contrast measures (i.e., difference in RMST, ratio of RMST, and ratio of RMTL). For the difference metric, the link function $g(\cdot)$ in the model above is the identity link. For the ratio metrics, the log-link is used. Specifically, with this pbc example, we are now trying to fit data to the following regression models:

1. Difference in RMST $$E(Y \mid arm,\ X) = \alpha + \beta (arm) + \gamma_1 (age) + \gamma_2(bili) + \gamma_3(albumin),$$
2. Ratio of RMST $$\log { E(Y \mid arm, \ X) } = \alpha + \beta (arm) + \gamma_1 (age) + \gamma_2(bili) + \gamma_3(albumin),$$
3. Ratio of RMTL $$\log { \tau - E(Y \mid arm, \ X) } = \alpha + \beta (arm) + \gamma_1 (age) + \gamma_2(bili) + \gamma_3(albumin).$$

Below is the output that rmst2 returns for the adjusted analyses.

rmst2(time, status, arm, tau=10, covariates=x)

The first block of the output is a summary of the adjusted treatment effect. Subsequently, a summary for each of the three models are provided.

## 4 Conclusions

The issues of the hazard ratio have been discussed elsewhere and many alternatives have been proposed, but the hazard ratio approach is still routinely used. The restricted mean survival time is a robust and clinically interpretable summary measure of the survival time distribution. Unlike median survival time, it is estimable even under heavy censoring. There is a considerable body of methodological research about the restricted mean survival time as alternatives to the hazard ratio approach. However, it seems those methods have been rarely used in practice. A lack of user-friendly, well-documented program with clear examples would be a major obstacle for a new, alternative method to be used in practice. We hope this vignette and the presented survRM2 package will be helpful for clinical researchers to try moving beyond the comfort zone - the hazard ratio.

## References

[1] Hernan, M. A. (2010). The hazards of hazard ratios. Epidemiology (Cambridge, Mass) 21, 13-15.

[2] Uno, H., Claggett, B., Tian, L., Inoue, E., Gallo, P., Miyata, T., Schrag, D., Takeuchi, M., Uyama, Y., Zhao, L., Skali, H., Solomon, S., Jacobus, S., Hughes, M., Packer, M. & Wei, L.-J. (2014). Moving beyond the hazard ratio in quantifying the between-group difference in survival analysis. Journal of clinical oncology : official journal of the American Society of Clinical Oncology 32, 2380-2385.

[3] Miller, R. G. (1981). Survival Analysis. Wiley.

[4] Tian, L., Zhao, L. & Wei, L. J. (2014). Predicting the restricted mean event time with the subject's baseline covariates in survival analysis. Biostatistics 15, 222-233.

[5] Andersen, P. K., Hansen, M. G. & Klein, J. P. (2004). Regression analysis of restricted mean survival time based on pseudo-observations. Lifetime data analysis 10, 335-350.

[6] Klein, J. P., Gerster, M., Andersen, P. K., Tarima, S. & Perme, M. P. (2008). SAS and R functions to compute pseudo-values for censored data regression. Computer methods and programs in biomedicine 89, 289-300.

[7] Parner, E. T. & Andersen, P. K. (2010). Regression analysis of censored data using pseudo-observations. The Stata Journal 10(3), 408-422.

## Try the survRM2 package in your browser

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text:
# Does the Newtonian gravitational field have momentum analogous to the Poynting vector?

We can define the total energy of the electromagnetic field as:

$$\mathcal{E}_{EM}= \frac{1}{2} \int_V \left(\varepsilon_0\boldsymbol{E}^2+\frac{\boldsymbol{B}^2}{\mu_0}\right)dV$$

which satisfies the conservation law:

$$\frac{\partial\mathcal{E}_K}{\partial t} + \frac{\partial \mathcal{E}_{EM}}{\partial t} = 0$$

where $$\mathcal{E}_K$$ is the kinetic energy of the matter. In addition, we can define the electromagnetic momentum as the Poynting vector:

$$\boldsymbol{p} = \frac{1}{\mu_0}\boldsymbol{E}\times\boldsymbol{B}$$

For the newtonian gravitational field $$\boldsymbol{g}$$, it seems possible to define an gravitational energy as:

$$\mathcal{E}_{G}= \frac{1}{2} \int_V \frac{\boldsymbol{g}^2}{4\pi G}dV$$

satisfying the corresponding conservation law. However, it does not seem that from the field alone it is possible to construct a gravitational analogue to the Poynting vector Is there any intuitive idea why this would not be possible?

• In electrodynamics, the Poynting vector forms part of the canonical stress-energy tensor, specifically the component $T^{0i}$. I suspect (though it's been a while since I've done a similar calculation, so caveat lector) that the analogous quantities vanish identically in Newtonian gravity because $\partial \mathcal{L}/\partial \dot{\phi} = 0$, i.e., the Lagrangian does not depend on $\dot{\phi}$. Jul 4 at 14:26
• Yes, in Gravitoelectromagnetism, but with the 4D representation of space-time (GR): en.wikipedia.org/wiki/Gravitoelectromagnetism Jul 4 at 15:51

No, in Newtonian mechanics there is no gravitational momentum. Physically, the reason is that the gravitational field doesn't propagate: it responds instantaneously to changes in the matter distribution, so that there is no trackable movement of energy from one place to another.

Mathematically, this is related that there are no time derivatives of $$\phi$$ in the equations of motion or the Lagrangian, unlike in electromagnetism. This is because Newtonian gravity is really the very low speed limit of general relativity - it's like working in an approximation where charges move so slowly that the electric field is quasistatic and there's no magnetic field, and thus no Poynting vector.

• Is there a Poynting equivalent in relativity? Jul 5 at 3:33
• Concerning "trackable movement of energy": if I'm not mistaken, the $T_0 {}^i$ components of the canonical stress-energy tensor tell you about the fluxes of energy. And if you calculate them, they're non-zero: $T_0 {}^i = - \frac{1}{4\pi G} \dot{\phi} (\partial^i \phi)$. If so, saying that "there's no trackable movement of energy" is a bit misleading. Jul 5 at 11:36
• @BlueRaja-DannyPflughoeft Sort of. The closest analog is the Landau-Lifshitz pseudotensor. However, as its name implies, it's not a tensor - it's designed to vanish in an inertial frame. The issue is that defining the energy and momentum of the gravitational field is a tricky thing, because gravity defines the very spacetime which you want to use to define energy and momentum. You either get zero, or weird non-tensorial constructions like the LL pseudotensor. Jul 5 at 13:51
• @MichaelSeifert That's a good point. I'm also not completely sure on what to do with a non symmetric energy-momentum tensor - and also not completely sure on how to interpret an energy flux for a field that responds instantly to changes. Jul 5 at 13:54
• Even with instantaneous propagation, I think that the statement of energy conservation would still be that $$\int_\mathcal{V} \left( \frac{1}{4 \pi G} (\nabla \phi)^2 + \rho_m \right) d^3r + \oint_{\partial \mathcal{V}} \left( -\frac{1}{4 \pi G} \dot{\phi} \nabla \phi \right) \cdot d\vec{a} = 0,$$where $\rho_m$ is the mechanical energy density. In other words, if the energy density changes in some volume, there's a corresponding change in the surface term (instantaneously) that can be interpreted as a flux of energy in or out of the volume. Jul 5 at 14:55

The Lagrangian for Newtonian gravitation (ignoring the kinetic energy of the sources, which will not affect this argument) is $$\mathcal{L} = - \frac{1}{8 \pi G} (\nabla \phi)^2 - \phi \rho$$ where $$\phi$$ is the gravitational field and $$\rho$$ is the density of the source matter. It is an easy exercise to confirm that this yields the Newtonian gravity equation $$\nabla^2 \phi = 4 \pi G \rho$$.

The above Lagrangian is invariant under spatial and temporal translations, so it will have a set of associated Noether currents that form a canonical stress-energy tensor: $$T_\mu {}^\nu = \left( \frac{\delta \mathcal{L}}{\delta (\partial_\nu \phi)} \right) \partial_\mu \phi - \delta_\mu {}^\nu \mathcal{L}.$$ Specifically, the components $$T_i {}^0$$ correspond to the momentum density for any Lagrangian; and by definition, the integral of the Noether current over all of space, $$P_i = \iiint T_i {}^0 \, d^3 \vec{r},$$ is constant with respect to $$t$$. This is the general way in which momentum conservation manifests itself in field theories. In electrodynamics, the quantity $$T_i {}^0$$ is related to the Poynting vector, and the integral over $$T_i {}^0$$ over space does give a quantity that's conserved in time, which is the total momentum of the system (field momentum plus mechanical momentum.)

But in the Newtonian Gravity Lagrangian, $$\dot{\phi}$$ does not appear at all, and so we have $$T_i {}^0 = 0$$ for the part of the Lagrangian that depends on the fields. In other words, the analogue of the Poynting vector in Newtonian gravity vanishes identically, and the statement of momentum conservation that is guaranteed by Noether's theorem only involves the mechanical momentum of the sources.

• To the OP: please don't accept this until a couple of days have gone by, as I am not 100% sure of the argumentation here (particularly the interpretation of the components of the canonical stress-energy tensor, and which of $T_i {}^0$ or $T_0 {}^i$ is the momentum density and which is the energy flux in situations like this one where the canoncial stress-energy is not symmetric.) To everyone else: please poke holes in this argument if you see any. Jul 4 at 15:20

I will offer a tentative answer, although I think Michael Seifert's answer is actually excellent (although for an arbitrary system of masses it is impossible to construct a gravitational potential independent of time). On the one hand, if we derive the gravitational energy density equation, we have that:

$$\frac{\partial \epsilon_G}{\partial t} = \frac{\boldsymbol{g}}{4\pi G}\cdot \frac{\partial \boldsymbol{g}}{\partial t}, \tag{1}$$

On the other hand, the equation relating the field to the source $$\nabla\cdot\boldsymbol{g} = 4\pi G \rho_m$$ and the continuity equation imply that:

$$\frac{\partial (\nabla\cdot\boldsymbol{g})}{\partial t} = 4\pi G \frac{\partial \rho_m}{\partial t} = -4\pi G \nabla\cdot \boldsymbol{j}_m, \qquad \Rightarrow \frac{\partial \boldsymbol{g}}{\partial t} = -4\pi G \boldsymbol{j}_m, \tag{2}$$

By introducing this last result in $$(1)$$, we obtain:

$$\frac{\partial \epsilon_G}{\partial t} = -\boldsymbol{g}\cdot \boldsymbol{j}_m, \tag{3}$$

We can see that this part is cancelled out with kineteric energy of matter:

$$\frac{\partial \epsilon_m}{\partial t} = \rho_m\boldsymbol{g}\cdot\boldsymbol{v} = \boldsymbol{g}\cdot\boldsymbol{j}_m, \qquad \Rightarrow \frac{\partial \epsilon_G}{\partial t}+ \frac{\partial \epsilon_m}{\partial t} = 0, \tag{4}$$

This result contrasts with the electromagnetic case where we have that:

$$\frac{\partial \epsilon_{EM}}{\partial t}+ \frac{\partial \epsilon_m}{\partial t} = -(\boldsymbol{E}\cdot\boldsymbol{j}_q + \mu_0^{-1}\boldsymbol\nabla \cdot (\mathbf{E}\times\mathbf{B})) + (\boldsymbol{E}\cdot\boldsymbol{j}_q)$$

So, in general, we have:

$$\frac{\partial \epsilon_{EM}}{\partial t}+ \frac{\partial \epsilon_m}{\partial t} = -\frac{1}{\mu_0}\boldsymbol\nabla \cdot (\mathbf{E}\times\mathbf{B})) \neq 0, \tag{5}$$

Thus, the main difference seems to be that in newtonian gravity the flux of energy at each point of space is zero $$(4)$$, precisely because matter and the gravitational field are instantaneously readjusted, which is relativistically untenable.

NOTE: In equation $$(2)$$ the more general solution is:

$$\frac{\partial \boldsymbol{g}}{\partial t} = -4\pi G \boldsymbol{j}_m + k\ \nabla\times(4\pi G\rho_m\boldsymbol{g})$$

although I originally considered $$k = 0$$, however, if one takes $$k \neq 0$$, then it opens the possibility of constructing a gravitational Poynting vector as:

$$\boldsymbol{S}_G = \boldsymbol{p}_G = 4\pi k G \rho_m\boldsymbol{g}$$

That vector would only be nonzero if the gravitational field is NOT conservative, because it is not an irrotational field (this happens, for example, if $$\boldsymbol{g}$$ explicitly depends on time: $$\boldsymbol{g} \neq -\nabla V_g$$).

• I believe there is a flaw in your logic, in line 2, the curl of some arbitrary vector function is present . Since the divergence of the curl is zero. However if I remember correctly when playing around with this myself integral form, this term is dependant on v, I could be wrong though. Jul 4 at 19:21
• @jensenpaull I am aware that I arbitrarily took that possible vector equal to zero, but that in fact the most general solution is $\dot{\boldsymbol{g}} = -4\pi G \boldsymbol{j}_m + k \boldsymbol{\nabla}\times \boldsymbol{b}$, with $k\neq 0$, then $k\boldsymbol{b}$ would be a kind of "graviatory Poynting vector". Jul 4 at 19:44
• @jensepaull Infact, the election $\boldsymbol{b} = \rho\boldsymbol{g}$ and $k \propto G$ is dimensionally consistent. Jul 5 at 22:47
• Less of a poynting vector since its not in the form div [blank], the closed surface integral of this is always zero. Jul 6 at 5:58


text:
How to link the assembly accession with the chromosome accession for prokaryotic representative genomes
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0
Entering edit mode
2.9 years ago
wangdp123 ▴ 250

Dear colleague,

I am working on the analysis of prokaryotic genomes from NCBI genome database.

After opening the file, we could see one column named "Chromosome RefSeq". (e.g., NZ_AQXM00000000)

Each file has a name like "GCF_000834735.1_ASM83473v1_protein.faa.gz".

It is odd that the two datasets use different accession number system. In this case, how to identify if the genome of the proteins is annotated in the file "prok_representative_genomes.txt"？

My aim is to retrieve all the protein sequences for the genomes listed in the file "prok_representative_genomes.txt".

Thanks a lot,

Kind regards

Tom

NCBI prokaryotic genomes • 678 views
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Entering edit mode

Thanks,

Does anybody know why there are some 0 values in the "Genomes" column of the file prok_representative_genomes.txt? For example, "Acetobacter aceti".

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The README says it's manually updated. I tend to believe you'll see a more complete picture when using the prokaryotes.txt (computationally updated...)

1
Entering edit mode
2.9 years ago
vkkodali ★ 2.7k

In the same FTP path, there is another file prokaryotes.txt that can be helpful. Join the two files on the organism name (column 3 of prok_representative_genomes.txt and column 1 of prokaryotes.txt). Then, extract the path to the assembly folder on the FTP site (column 21 of prokaryotes.txt file) and download the *.protein.faa.gz files.

Accessions with the NZ_ prefix are RefSeq Chromosome accessions whereas the ones with GCF_ prefix are assembly accessions. An assembly includes both chromosome and plasmid(s).

0
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As vkkodali wrote, use the prokayotes.txt. However, there's no need to join, one can filter on the Reference column. The NCBI discriminates between manually selected reference (exist for common lab organisms) and computationally selected representative genomes (exists for each species, almost).


text:
# $(a_n)$ is a bounded sequence and every convergent subsequence of $(a_n)$ converges to $a\in\mathbb{R}$. Show $(a_{n})$ converges to $a$.

To me, the problem is asking:

Assume $(a_n)$ is a bounded sequence. Prove:$[\text{Every convergent subsequence of }(a_n)\text{ converges to }a]\implies[\lim a_n=a]$

Step 0: We decide we'll prove the contrapositive, so the problem is now:

Assume $(a_n)$ is a bounded sequence. Prove: $[\lim a_n\ne a]\implies[\text{There exists a convergent subsequence of }(a_n)\text{ that does not converge to }a ]$

Step 1: $\lim a_n\ne a\implies\exists\ \epsilon_{0}>0:|a_n-a|\ge \epsilon_{0}\ \forall\ n\in\mathbb{N}$

Step 2: $(a_n)$ is bounded $\implies$ $(a_n)$ has a convergent subsequence. Call this convergent subsequence $(a_{n_{k}})$.

Step 3: The only things we know about $(a_{n_{k}})$ are that A) it converges to something and B) it's a subsequence of $(a_n)$. I think the only task left is to show $(a_{n_{k}})$ cannot converge specifically to $a$.

Step 4: To show $(a_{n_{k}})$ does not converge to $a$, we must show $\exists\ \epsilon_{1}>0:|a_{n_{k}}-a|\ge\epsilon_{1}\ \forall\ k\in\mathbb{N}$.

Step 5: $(a_{n_{k}})$ is a subsequence of $(a_n)$ $\implies$ All the terms of $(a_{n_{k}})$ come from $(a_{n})$. Since all terms of $(a_{n})$ are at least $\epsilon_{0}$ away from $a$, then so are all terms of $(a_{n_{k}})$.

Step 6: Let $\epsilon_{1}=\epsilon_{0}$. Then we have found an $\epsilon_{1}$ such that $|a_{n_{k}}-a|\ge \epsilon_{1}\ \forall\ k\in\mathbb{N}$.

Conclusion: $(a_{n_{k}})$ is a convergent subsequence of $(a_{n})$ (step 2) that does not converge to $a$ (steps 4-6). So we've found a convergent subsequence of $(a_{n})$ that doesn't converge to $a$.

I wrote this in these steps so it's easier to point out where I'm wrong. I don't understand why every solution I read (that doesn't involve limit superior) requires a sub-subsequence. It seems to me the reason the convergent sub-subsequence doesn't converge to $a$ is the same reason the convergent subsequence $(a_{n_{k}})$ doesn't converge to $a$, so why do we have to make a sub-subsequence?

I know this exact question has been asked on here ad nauseam. I read the posts I could find, but I still don't understand why a sub-subsequence is necessary. I'm self-learning, and I've been stuck on this for the past three days. I decided to finally just ask. I'm sorry.

• Do note that $(a_n)$ is a subsequence of $(a_n)$. Aug 13 '17 at 13:44
• Well, step 1 is already wrong. we could have $a_{2n}=a\,\forall n$ for example without that showing that $\lim a_n=a$.
– lulu
Aug 13 '17 at 13:48
• Counter-example: let $a_n =n$. The left hand side of the implication is vacuously true for all $a$. Aug 19 '17 at 20:58

Your proof is wrong, but it can be easily fixed. In step one, argue that there is a subsequence $(a_{n_k})_k$ of $(a_n)_n$ for which the conclusion in step $1$ holds. Then this subsequence will be bounded, so we can find a subsequence of this subsequence and make the same conclusions you did.

The reason this sort of argument requires a further subsequence is that just knowing that $(a_n)_n$ doesn't converge to $a$ means only that infinitely many terms of $a_n$ are away from $a$. However, in order to proceed with the proof, we need to have all terms away from $a$. Thus we need to take a subsequence first before proceeding with the proof - which will require a further subsequence.

In fact, something important regarding sequential convergence on topological spaces (or in particular, the real line) is at work here.

A sequence converges to $x$ iff every subsequence has a further subsequence that converges to $x$.

Proof: The forward direction is immediate. For the converse, suppose $x_n\not\to x$. Then there is a subsequence $(x_{n_k})_k$ of $(x_n)_n$ that stays $\varepsilon_0$ away from $x$. Hence no subsequence of $(x_{n_k})_k$ converges to $x$.

• Oh I see, I've been misunderstanding the negation of $\lim a_n =1$ from the start. I didn't realise there could still be a few terms of $(a_n)$ close to $a$ (I thought $\lim a_{n}\ne a$ implied ALL of $(a_{n})$'s terms are $\epsilon_{0}$ away.) Thank you so much for help. Aug 13 '17 at 14:38
• @anonanon444 A simple example to illustrate why this fails is the sequence $((-1)^n)_n$. This sequence doesn't converge to $1$ but it has infinitely many terms equal to $1$. Aug 13 '17 at 15:15

You can do without subsequences at all.

There is an $R\geq0$ such that $A:=\{a_n| n\geq1\}\subset B_R:=[{-R},R]$.

Let a neighborhood $U_\epsilon$ of $a$ be given. By assumption on the sequence $(a_n)_{n\geq1}$, for all $x\in B_R\setminus\{a\}$ there is a neighborhood $U_\delta(x)$ ($\delta$ may depend on $x$) such that $U_\delta(x)\cap A\subset\{x\}$. The family $\{U_\epsilon(a)\}\cup \bigl\{U_\delta(x) \bigm| x\in B_R\setminus\{a\}\bigr\}$ is an open covering of $B_R$, hence will contain a finite subcovering. There will be only finitely many $U_\delta(x_i)$ in this subcovering, hence there is an $n_0$ such that all $a_n$ with $n>n_0$ are lying in $U_\epsilon(a)$.


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## OOP

Gradual Typing for Objects. Jeremy Siek and Walid Taha.

Static and dynamic type systems have well-known strengths and weaknesses. In previous work we developed a gradual type system for a functional calculus named [...]. Gradual typing provides the benefits of both static and dynamic checking in a single language by allowing the programmer to control whether a portion of the program is type checked at compile-time or run-time by adding or removing type annotations on variables. Several object-oriented scripting languages are preparing to add static checking. To support that work this paper develops [another calculus], a gradual type system for object-based languages, extending the [...] calculus of Abadi and Cardelli. Our primary contribution is to show that gradual typing and subtyping are orthogonal and can be combined in a principled fashion. We also develop a small-step semantics, provide a machine-checked proof of type safety, and improve the space efficiency of higher-order casts.

The authors' previous work on gradual typing was discussed here. This brings it to an object-oriented setting which is (as the abstract points out) very directly applicable to mainstream scripting languages, at least in principle.

[Edit: This is from the types list, where the authors also added: "We will present the paper at ECOOP 2007 and would be especially interested in any feedback on the paper before the final submission is due on April 25."]

## First Class Relationships in an Object-oriented Language

First Class Relationships in an Object-oriented Language, by Gavin Bierman and Alisdair Wren, was a paper published at ECOOP 2005.

They show how to add relationships as a first-class mechanism to a Java-like language, where by relationships in something like the UML sense. Cribbing from their examples, you might have a Student class and a Course class, and an Attends relationship, which is a binary relation between Students and Courses. Then, there are mechanisms to dynamically add and remove entries from a relationship, and to query a relationship about whether particular objects are in them or not.

Now my personal random editorialisation: I think this is a really interesting idea, because these kinds of things pops up all the time in OO models, and having relations be first class means that you can move some state out of individual object instances, which is always a good thing, and the type system can guarantee something about what relationships will hold. The things that scare me about this idea are first, that you can potentially add a lot of heap pressure by keeping objects live longer, and second, that you now have much more pervasive aliasing of objects in your program. I don't know if these are real problems though, and anyway the feature is cool enough that I'd be interested in programming in a language with support for it, just to see what it's like.

## Directness and liveness in the morphic user interface construction environment

Directness and liveness in the morphic user interface construction environment, John H. Maloney and Randall B. Smith, 1995.

Morphic is a user interface construction environment that strives to embody directness and liveness. Directness means a user interface designer can initiate the process of examining or changing the attributes, structure, and behavior of user interface components by pointing at their graphical representations directly. Liveness means the user interface is always active and reactive-objects respond to user actions, animations run, layout happens, and information displays update continuously. Four implementation techniques work together to support directness and liveness in Morphic: structural reification, layout reification, ubiquitous animation, and live editing.

Morphic was developed in Self and then adopted by Squeak. Reading this paper makes Squeak more interesting to click around in!

## Interesting project to modularize Squeak

Ralph Johnson mentions an interesting project from Pavel Krivanek to modularize Squeak. For example, there's a KernelImage that excludes the GUI (it's 2.8 MB compared to 15 MB for the full 3.9 release). The modularized images are created automatically from the complete image, by Squeak code.

It's been awhile since we discussed Squeak, which remains as far as I can tell, an interesting project worth keeping an eye on.

## Is "post OO" just over?

While studying the conference program of the upcoming OOPSLA 2006 I discovered under the category "essay" an author who has quite something critical to say about AOP:

Aspect-oriented programming is discussed as a promising new technology. Like object-oriented programming, it is beginning to pervade all areas of software engineering. With its growing popularity, practitioners and academics alike are beginning to wonder whether they should start looking into or it, or otherwise risk having missed an important development. The author of this essay finds that much of aspect-oriented programming's success seems to be based on the conception that it improves both modularity and the structure of code, while in fact, it actually works against the primary purposes of the two, namely independent development and understandability of programs. Not seeing any way of fixing this situation, he thinks the success of aspect-oriented programming to be paradoxical.

This is not just another internet rant about the latest PL hype but the author, Friedrich Steimann, had done interesting work about AOP before. In particular his latest paper about typed AOP:

AOP and the antinomy of the liar

but also his award winning former critical AOP review:

Domain models are aspect free

## Revealing the X/O impedance mismatch

Ralf LÃ¤mmel and Erik Meijer. Revealing the X/O impedance mismatch.

When engaging in X/O mapping, i.e., when viewing XML data as objects or vice versa, one faces various conceptual and technical challenges -- they may be collectively referred to as the X/O impedance mismatch'. There are these groups of challenges. (i) The XML data model and the object data model differ considerably. (ii) The native XML programming model differs from the normal OO programming model. (iii) The typical type systems for XML and objects differ considerably, too. (iv) Some of the differences between data models and between type systems are more like idiosyncrasies on the XML site that put additional burden on any X/O mapping effort. (v) In some cases, one could ask for slightly more, not necessarily XML-biased language expressiveness that would improve X/O mapping results or simplify efforts.

The present article systematically investigates the mismatch in depth. It identifies and categorizes the various challenges. Known mitigation techniques are documented and assessed; several original techniques are proposed. The article is somewhat biased in that it focuses on XML Schema as the XML type system of choice and on C# as the OO programming language of choice. In fact, XSD and C# are covered quite deeply. Hence, the present article qualifies as a language tutorial of the other kind'.

This paper is over 100 pages, way longer than I have the time to read at the moment. Skimming, the paper looks interesting and useful. If you manage to read the whole thing, do share your observations with us in the discussion group.

## Securing the .NET Programming Model

Securing the .NET Programming Model. Andrew J. Kennedy.

The security of the .NET programming model is studied from the standpoint of fully abstract compilation of C#. A number of failures of full abstraction are identified, and fixes described. The most serious problems have recently been fixed for version 2.0 of the .NET Common Language Runtime.

This is highly amusing stuff, of course. Some choice quotes:

if source-language compilation is not fully abstract, then there exist contexts (think â€˜attackersâ€™) in the target language that can observably distinguish two program fragments not distinguishable by source contexts. Such abstraction holes can sometimes be turned into security holes: if the author of a library has reasoned about the behaviour of his code by considering only source-level contexts (i.e. other components written in the same source language), then it may be possible to construct a component in the target language which provokes unexpected and damaging behaviour.

One could argue that full abstraction is just a nicety; programmers donâ€™t really reason about observations, program contexts, and all that, do they? Well, actually, I would like to argue that they do. At least, expert programmers...

"A C# programmer can reason about the security properties of component A by considering the behaviour of another component B written in C# that â€œattacksâ€ A through its public API." -
This can only be achieved if compilation is fully abstract.

To see the six problems identified by thinking about full abstraction you'll have to go read the paper...

## Variance and Generalized Constraints for C# Generics

Variance and Generalized Constraints for C# Generics. Burak Emir, Andrew J. Kennedy, Claudio Russo, Dachuan Yu. July 2006

Generic types in C# behave invariantly with respect to sub-typing. We propose a system of type-safe variance for C# that supports the declaration of covariant and contravariant type parameters on generic types. To support more widespread application of variance we also generalize the existing constraint mechanism with arbitrary subtype assertions on classes and methods. This extension is useful even in the absence of variance, and subsumes equational constraints proposed for Generalized Algebraic Data Types (GADTs). We formalize the subtype relation in both declarative and syntax-directed style, and describe and prove the correctness of algorithms for constraint closure and subtyping. Finally, we formalize and prove a type safety theorem for a featherweight language with variant classes and generalized constraints.

Discussion of previous C# GADT paper on LtU.

I am unsure about use-site versus definition-site variance declerations. It would be interesting to hear what others think.

Also check out the LtU discussion on wildcards in Java.

## Class decorators in Python

Guido resisted the few calling for class decorators, because there wasn't a clear use case that wasn't more readable done another way... [but] Guido has conceded, class decorators will make it into some future version of Python.


text:
# Problem of the Week #287 - Apr 09, 2019

Status
Not open for further replies.

#### Euge

##### MHB Global Moderator
Staff member
Here is this week's POTW:

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Let $X$ be a locally compact Hausdorff space, and let $\mu$ be a Radon measure on $X$. Recall that the complement of the support of $\mu$ is the union of all open subsets of $X$ of $\mu$-measure zero. Show that the support of $\mu$ is the set of all $x\in X$ such that for all compactly supported continuous functions $f : X\to [0,1]$ with $f(x) > 0$, the integral $\int_X f\, d\mu > 0$.

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text:
## Thursday, October 26, 2006

### Estimating the liability of state governments

A new RBI committee report on liabilities of State governments is now on the RBI web site. It has some interesting new material. But they seem to have missed out on some pieces of literature survey. E.g. the committee does not seem to know about Bhardwaj & Dave, 2005, the best work so far on estimating the liabilities on account of pensions.

While I'm on this subject, I'd like to draw your attention to a summary table of the liabilities of the US government done by the GAO. It's a great table - I would be thrilled when we are able to do such a table for India - I wonder what is the agency in India which can become such a GAO? It's interesting to see that they fold their liabilities on account of civil servants health and pensions into the early part of the table. On a related note, there's a great article in The Washington Post by Matt Crenson titled GAO Chief Warns Economic Disaster Looms.

In the Indian case, in estimating the liabilities of the State, we'd have to add in the liabilities on account of the implicit put option that the State has out on all public sector finance companies. Susan Thomas and I did some measurement work on this question in 2000, with a paper titled Systemic fragility in Indian banking: Harnessing information from the equity market. In the following years, the idea of using this option theory approach has become much more mainstream (e.g. papers by Luc Laeven etc). In the years that followed, the moneyness of the put worked out fine, but that does not change our measurement of the contingent liability faced by the Indian State circa 2000. There is a new NBER paper by Gray, Merton and Bodie on a related theme.

Sometimes, it is felt that in the case of US-64, the political process reached a roughly 50-50 allocation of the pain, between citizens and taxpayers. Does this bode well for the future? I don't believe so. In the case of US-64, there was no legal liability on the hands of the government - even if UTI claimed the units were worth Rs.14 while they were really worth Rs.6, there was no legal obligation upon GOI to pay customers anything. In contrast, when it came to the assured return schemes, there was no space for GOI to dodge any of the liability that UTI had written down in black and white. In similar fashion, there is no space for GOI to avoid the liabilities of PSU banks and insurance companies. So in short, the fact that GOI did not go up to Rs.14 for US-64 does not change anything in terms of the contingent liability that GOI faces owing to other public sector finance companies.

1. "...the political process reached a roughly 50-50 allocation of the pain, between citizens and taxpayers."

That's a funny statement. But it says a lot about how the politics of Indian economy work.

2. Ideally CAG would be poised to take on the functons of GAO, but only ideally speaking!

3. Yes, a CAG on steroids would become the GAO. But from what little I know about the two agencies, there is a big gap between the two right now.

The 50-50 pain sharing is a fair description of what happened on US-64. I personally think this outcome was quite a bit better than what would have come out of the ordinary pessimistic portrayal of the Indian State.

Please note: Comments are moderated; I will delete comments that misbehave. The rules are as follows. Only civilised conversation is permitted on this blog. Criticising me is perfectly okay; uncivilised language is not. I delete any comment which is spam, has personal attacks against anyone, or uses foul language.

Please note: LaTeX mathematics works. This means that if you want to say $10 you have to say \$10.


text:
# In Exereises 24-29, solve by Cramer s rule; where it applies24. Tx1 _ 2x2 = 3 3x1 + Xz =

###### Question:

In Exereises 24-29, solve by Cramer s rule; where it applies 24. Tx1 _ 2x2 = 3 3x1 + Xz = 5 25. 4x + Sy =2 Hx + Y + 2z = 3 x + Sy + 2z = ]

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