4,998 results
Search Results
2. Entropy criteria and stability of extreme shocks: a remark on a paper of Leger and Vasseur
- Author
-
Kevin Zumbrun and Benjamin Texier
- Subjects
Conservation law ,Kullback–Leibler divergence ,Standard molar entropy ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Regular polygon ,Min entropy ,Shock strength ,01 natural sciences ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Uniqueness ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations, Comment: to appear in Proceedings of the AMS
- Published
- 2014
3. Class groups in cyclic $\ell $-extensions: Comments on a paper by G. Cornell
- Author
-
Michael Rosen
- Subjects
Algebra ,Class (set theory) ,Applied Mathematics ,General Mathematics ,Mathematics - Published
- 2013
4. Remarks on a paper by Chao-Ping Chen and Feng Qi
- Author
-
Koumandos, S. and Koumandos, S. [0000-0002-3399-7471]
- Subjects
Discrete mathematics ,Ping (video games) ,Monotonicity ,Sequence ,biology ,Applied Mathematics ,General Mathematics ,Best bounds ,Monotonic function ,biology.organism_classification ,Upper and lower bounds ,Chen ,Gamma function ,Calculus ,Wallis' inequality ,Special case ,Mathematics - Abstract
In a recent paper, Chao-Ping Chen and Feng Qi (2005) established sharp upper and lower bounds for the sequence P n := 1.3 … ( 2 n − 1 ) 2.4 … 2 n P_{n}:=\frac {1.3\ldots (2n-1)}{2.4\ldots 2n} . We show that their result follows easily from a theorem of G. N Watson published in 1959. We also show that the main result of Chen and Qi’s paper is a special case of a more general inequality which admits a very short proof.
- Published
- 2005
5. A weighted uniform $L^{p}$--estimate of Bessel functions: A note on a paper of Guo
- Author
-
Krzysztof Stempak
- Subjects
symbols.namesake ,Cylindrical harmonics ,Bessel process ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Struve function ,Bessel polynomials ,symbols ,Calculus ,Bessel function ,Lommel function ,Mathematics - Published
- 2000
6. Remarks on DiPerna’s paper 'Convergence of the viscosity method for isentropic gas dynamics'
- Author
-
Gui-Qiang Chen
- Subjects
Discrete mathematics ,Isentropic process ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Vacuum state ,Finite difference method ,Euler equations ,Binary entropy function ,symbols.namesake ,Riemann hypothesis ,Compact space ,Mathematics Subject Classification ,symbols ,Mathematics - Abstract
Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.
- Published
- 1997
7. A remark on a paper of E. B. Davies
- Author
-
Maciej Zworski
- Subjects
Applied Mathematics ,General Mathematics ,Mathematics - Published
- 2001
8. A remark on a paper by C. Fefferman: 'The uncertainty principle' [Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206; MR0707957 (85f:35001)]
- Author
-
Filippo Chiarenza and Michele Frasca
- Subjects
Uncertainty principle ,Applied Mathematics ,General Mathematics ,Calculus ,Mathematics - Abstract
We give a simplified proof of an imbedding theorem by C. Fefferman [3].
- Published
- 1990
9. Noncomplete linear systems on elliptic curves and Abelian varieties: Addendum to a paper by Ch. Birkenhake
- Author
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E. Ballico
- Subjects
Pure mathematics ,Elliptic curve ,Applied Mathematics ,General Mathematics ,Linear system ,Addendum ,Abelian group ,Algorithm ,Mathematics - Abstract
Here we give a result on the postulation (i.e. the 2-normality) of nonlinearly normal embeddings of Abelian varieties. This result improves some of the results proved in a recent paper by Ch. Birkenhake.
- Published
- 1998
10. Remark on a paper of Cheng and Smoller
- Author
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Vinod B. Goyal
- Subjects
Singularity ,Differential equation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Isolated singularity ,Mathematics ,Mathematical physics - Abstract
A class of functions K(r) and the corresponding solutions are produced in the case of the differential equation d 2 u/dr 2 +1 du/r dr+K(r)e 2u(r) =1. The behavior of these solutions at an isolated singularity is also investigated
- Published
- 1991
11. A remark on a paper of J. S. Ruan: 'Invariant subspace of strictly singular operators' [Proc. Amer. Math. Soc. 108 (1990), no. 4, 931–936; MR1002160 (90g:47009)]
- Author
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Patrick M. Fitzpatrick and Seymour Goldberg
- Subjects
Discrete mathematics ,Algebra ,Applied Mathematics ,General Mathematics ,Invariant subspace ,Finite-rank operator ,Reflexive operator algebra ,Operator norm ,Invariant subspace problem ,Strictly singular operator ,Mathematics ,Bounded operator - Abstract
We observe that a strictly singular operator is not necessarily condensing, so that the invariant subspace problem for strictly singular operators remains open.
- Published
- 1991
12. Some remarks on a paper by R. H. Bruck
- Author
-
Trevor Evans
- Subjects
Loop (topology) ,Pure mathematics ,Rank (linear algebra) ,Subvariety ,Applied Mathematics ,General Mathematics ,Variety (universal algebra) ,Ring of integers ,Identity (music) ,Mathematics - Abstract
Introduction. In a recent paper [2] R. H. Bruck has introduced the concept of right neoring and discussed some properties of these systems. In particular, he has considered analogues of certain properties of the ring of integers. This paper is essentially a commentary on Bruck's paper and we generalize some of his results as follows. The construction of the universal right neoring in [2] is applied to the free monogenic $3-loop in any subvariety $3 of the variety of loops and a complete analogue of Theorem 4.1 of [2] is obtained for any one of these subvarieties. Then, using a result similar to those obtained in [5], it is shown that this construction yields uncountably many right neorings with an identity which generates the additive loop of the right neoring. Conversely, every right neoring with an identity which generates its additive loop can be obtained from a free monogenic $3-loop by the above construction. Each of these right neorings has some properties resembling those of the ring of integers. One possible answer is given to the question raised by Bruck concerning the existence of universal right neorings with free additive loop of arbitrary rank. A brief proof is given, using the results of [4; 5], of the cancellation properties of the monogenic universal right neoring. Finally, we discuss briefly the relationship between right neorings and the logarithmetics of Etherington.
- Published
- 1956
13. Direct product of division rings and a paper of Abian
- Author
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M. Chacron
- Subjects
Subdirect product ,Nilpotent ,Ring (mathematics) ,Pure mathematics ,Noncommutative ring ,Applied Mathematics ,General Mathematics ,Order (ring theory) ,Von Neumann regular ring ,Commutative property ,Direct product ,Mathematics - Abstract
It is shown that the rings under the title admit an order-theoretical characterization as in the commutative case studied by Abian. Introduction. Let R be an associative ring equipped with the binary relation (^) defined by xay if and only if xy = x2 in R. In this paper, it is shown that ( ^ ) is an order relation on R if and only if, R has no nilpotent elements i9*0). Conditions on the binary relation (g) in order that R split into a direct product of division rings are then studied in the light of Abian's result (l, Theorem l). Using similar argumentation and using certain subdirect representation of rings with no nilpotent elements, one obtains a complete similarity with the commutative case (yet, no extra complication in the computa- tions). Conventions. R is an associative ring which is, unless otherwise stated, with no nilpotent elements (other than 0). As a result of (2), R can be embedded into a direct product of skewdomains, R—* YLiei £i (that is to say, rings R, having no one-sided divisors of zero). The former embedding is fixed throughout the paper. It is therefore legiti- mate to identify any element x in R with the family consisting of all its projections (xj.e/. Finally, all definitions in (l) are extended (verbatim) to the present case (of a noncommutative ring R) and are freely used throughout.
- Published
- 1971
14. An operator valued function space integral: A sequel to Cameron and Storvick’s paper
- Author
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D. L. Skoug and G. W. Johnson
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Multiple integral ,Integral representation theorem for classical Wiener space ,Mathematical analysis ,Riemann integral ,Riemann–Stieltjes integral ,Singular integral ,Fourier integral operator ,Volume integral ,symbols.namesake ,symbols ,Daniell integral ,Mathematics - Abstract
Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.
- Published
- 1971
15. Extension of invariant linear functionals: a sequel to Fan’s paper
- Author
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Anthony Toming Lau
- Subjects
Almost periodic function ,Semigroup ,Computer Science::Information Retrieval ,Applied Mathematics ,General Mathematics ,Locally convex topological vector space ,Mathematical analysis ,Invariant (mathematics) ,Mathematics - Abstract
We establish the relationship between certain invariant extension properties for linear functional of K. Fan when a semigroup S acts on a locally convex topological vector space and the existence of a left-invariant mean on the space of almost periodic and weakly almost periodic functions on S.
- Published
- 1977
16. Corrections to the paper 'Remarks on fluctuations of sums of independent random variables'
- Author
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M. Kac and K. L. Chung
- Subjects
Exchangeable random variables ,Pairwise independence ,Independent and identically distributed random variables ,Convergence of random variables ,Multivariate random variable ,Applied Mathematics ,General Mathematics ,Sum of normally distributed random variables ,Statistical physics ,Algebra of random variables ,Central limit theorem ,Mathematics - Published
- 1953
17. A correction to the paper: 'Semi-open sets and semi-continuity in topological spaces' (Amer. Math. Monthly 70 (1963), 36–41) by Norman Levine
- Author
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T. R. Hamlett
- Subjects
Algebra ,Semi-continuity ,Applied Mathematics ,General Mathematics ,Calculus ,Semi open ,Topological space ,Mathematics - Abstract
A subset A A of a topological space is said to be semi-open if there exists an open set U U such that U ⊆ A ⊆ Cl ( U ) U \subseteq A \subseteq \operatorname {Cl} (U) where Cl ( U ) \operatorname {Cl} (U) denotes the closure of U U . The class of semi-open sets of a given topological space ( X , T ) (X,\mathcal {T}) is denoted S .O . ( X , T ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) . In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if T \mathcal {T} and T ∗ {\mathcal {T}^ \ast } are two topologies for a set X X such that S .O . ( X , T ) ⊆ S .O . ( X , T ∗ ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) \subseteq {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast }) , then T ⊆ T ∗ \mathcal {T} \subseteq {\mathcal {T}^ \ast } . In a corollary to this theorem, Levine states that if S .O . ( X , T ) = S .O . ( X , T ∗ ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) = {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast }) , then T = T ∗ \mathcal {T} = {\mathcal {T}^ \ast } . An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subsets can arise from functions, and points out some of the implications of two topologies being related in this manner.
- Published
- 1975
18. A note concerning Fox’s paper on Fenchel’s conjecture
- Author
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T. C. Chau
- Subjects
Combinatorics ,Normal subgroup ,Fuchsian group ,Conjecture ,Integer ,Applied Mathematics ,General Mathematics ,Lemma (logic) ,Calculus ,Order (group theory) ,Direct proof ,Finitely-generated abelian group ,Mathematics - Abstract
The object of this note is to point out and correct an error in the paper [21 of Fox, purporting to prove Fenchel's conjecture that a finitely generated. infinite Fuchsian group has a torsion-free normal subgroup of finite index. Fox divided the proof of the main result in his paper into four cases and an error occurs in the proofs of Cases (III) and (IV). A direct proof of Case (III) was given. While a direct proof of Case (IV) can be given, the author shows that it also follows indirectly from the result in the paper [1] of Burns and Solitar. Given any three integers a > 1, b > 1 and c > 1, there exist permutations A of order a and B of order b, such that AB has order c. Fox divided the proof of this lemma into the four cases: (I) ab 0 (mod 2) and max{a, b} ? c < a + b, (II) ab 0 (mod 2) and c : a + b, (III) a b c 1 (mod 2), (IV) a b 1 and c 0 (mod 2), and subcases. In the subcase m(a + b 2) + 2 ? c < (m + l)(b + 1) 1 of Case (III) where m is the largest integer not exceeding (c l)/(b + 1), and it is assumed that a ? b < c, the number of distinct symbols in the permutations [2, p. 64] nl~~~W B ( Wv-sI .. WI2 I7 (vWb B = (vl * *t*v W_ ***W P2) * (V W2 .. * ]Pi+]1)'
- Published
- 1983
19. On a paper of Reich concerning minimal slit domains
- Author
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James A. Jenkins
- Subjects
Discrete mathematics ,Compact space ,Polymer science ,Projection (mathematics) ,Applied Mathematics ,General Mathematics ,Nowhere dense set ,Point (geometry) ,Limit (mathematics) ,Measure (mathematics) ,Domain (mathematical analysis) ,Mathematics ,Complement (set theory) - Abstract
1. In a recent paper [2] Reich has made the observation that an example of Koebe given in 1918 [1] does not fulfill its asserted purpose. This example was to show that vanishing measure of the complement did not assure that a slit domain was minimal. Reich proceeded to fill the gap by carrying out the following somewhat more general construction. Let A be a compact perfect nowhere dense set on the x-axis in the z-plane (z=x+iy). Then there exists a compact set S in the z-plane with the properties: (i) A is the projection of S on the x-axis, (ii) S is composed of segments symmetric with respect to the x-axis and points on the x-axis, at least one segment being present, (iii) any point in S, not on the x-axis and not at the end of a segment in S, is the limit, both from the left and right, of points of S. Once this construction is performed the desired examples are easily given [2, ?4]. The object of the present paper is to give an alternative construction which is very explicit and direct.
- Published
- 1962
20. Remarks on a paper by A. Aziz
- Author
-
Zalman Rubinstein
- Subjects
Applied Mathematics ,General Mathematics ,Mathematics - Abstract
The paper consists of two parts. In the first part a short proof of the main theorem due to A. Aziz on the location of zeros of composite polynomials is given. In the second part some properties of a fixed length continuation of a polynomial are deduced.
- Published
- 1985
21. On a paper of Phelps
- Author
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Robert Sine
- Subjects
Unit sphere ,Combinatorics ,Convex hull ,Dense set ,Applied Mathematics ,General Mathematics ,Retract ,Function (mathematics) ,Extreme point ,Disk algebra ,Continuous functions on a compact Hausdorff space ,Mathematics - Abstract
In [4], Phelps showed that in certain function algebras the unit ball is the closed convex hull of its extreme points. The algebra, C(X), of complex valued continuous functions on a compact Hausdorff space, will always have this property. The class of logmodular algebras which have a Gleason part which is total also was shown to have the property. In this paper we give an elementary proof of the first result (a proof which is, in theory at least, constructive). The simplest nontrivial example of a logmodular algebra with a total part is the disk algebra (i.e. the functions continuous on the closed disk and analytic in the interior). For this algebra we show that the extreme points (in fact the exposed points) of the unit ball form a dense subset of the boundary of the unit ball. Let U be the unit ball of C(X). It is well known that q in U is an extreme point of U if I q(x) I = 1 for all x in X. Now if f in U never vanishes, then f is in the closed convex hull of the extreme points; in factf is between two uniquely determined extreme points. We need only observe that for each x in X,f(x) is halfway between two uniquely determined extreme points of the disk and that these points vary continuously with x. Now it is not necessarily the case that each function in U can be approximated by a nonvanishing function. For a counterexample we need only look at h(z) = z in the algebra of continuous functions on the disk. (If If(z)-zj 1/e. Letfi, i = 1, *, N be the retract maps obtained from these rotations. Then I 1/N Efi(z)-z
- Published
- 1967
22. Note on a paper by Shepperd on the braid group
- Author
-
Seymour Lipschutz
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,Braid group ,Braid ,Decision problem ,Automorphism ,Mathematics - Abstract
where the oi are the usual generators of Bn. (See [1; 4].) A textile manufacturer asked the following question: Can one decide whether or not a braid in An is also in Qn? (It is possible to "weave" a braid in Qn while keeping its ends "tied together.") Shepperd [6] solved the above decision problem using the generators and relations of the braid group and subgroups. In this paper we give another distinct solution working in terms of automorphisms of free groups. In particular, we use formulas developed in [3; 4]. The author wishes to thank his teacher, and friend, Professor Wilhelm Magnus who has contributed more than anyone else towards the author's knowledge of mathematics.
- Published
- 1963
23. A note on a paper of J. D. Stein, Jr.: 'Sequence of regular finitely additive set functions' (Trans. Amer. Math. Soc. 192 (1974), 59–66)
- Author
-
Surjit Singh Khurana
- Subjects
Discrete mathematics ,Set function ,Applied Mathematics ,General Mathematics ,Arithmetic ,Sequence (medicine) ,Mathematics - Abstract
Among other results, it is proved that if a sequence { μ n } \{ {\mu _n}\} of regular measures on a Hausdorff space, with values in a normed group, is convergent to zero for all σ \sigma -compact sets or all open sets, then there exists a maximal open set U such that μ ˙ n ( U ) → 0 , { μ ˙ n } {\dot \mu _n}(U) \to 0,\{ {\dot \mu _n}\} being the associated submeasures.
- Published
- 1977
24. Remark on a paper of Y. Ikebe
- Author
-
Ivan Singer
- Subjects
Applied Mathematics ,General Mathematics ,Mathematics - Published
- 1969
25. Remarks on a paper by Utz
- Author
-
George Brauer
- Subjects
Applied Mathematics ,General Mathematics ,Mathematics - Published
- 1958
26. A note on D. Quillen’s paper: 'Projective modules over polynomial rings' (Invent. Math. 36 (1976), 167–171)
- Author
-
Moshe Roitman
- Subjects
Discrete mathematics ,Collineation ,Applied Mathematics ,General Mathematics ,Complex projective space ,Polynomial ring ,Projective line over a ring ,Projective cover ,Projective space ,Projective module ,Quaternionic projective space ,Mathematics - Abstract
We give a simplified proof to the following theorem due to D. Quillen: if A is a commutative noetherian ring of global dimension ⩽ 1 \leqslant 1 , then finitely generated projective modules over A [ T 1 , … , T n ] A[{T_1}, \ldots ,{T_n}] are extended from A. We prove also that if A is a commutative noetherian ring of global dimension d, then finitely generated projective modules of rank > d > d over A [ T 1 , … , T n ] A[{T_1}, \ldots ,{T_n}] are extended from A.
- Published
- 1977
27. A note on a paper of Paul Hill and Charles Megibben
- Author
-
Joel Winthrop and Doyle O. Cutler
- Subjects
Combinatorics ,Lemma (mathematics) ,Cardinality ,Pure subgroup ,Selection (relational algebra) ,Statement (logic) ,Applied Mathematics ,General Mathematics ,Existential quantification ,Linear independence ,Cardinality of the continuum ,Mathematics - Abstract
This note deals with the correction of a small flaw in the proof of a theorem (and a lemma) in [2]. Also a simpler proof of the theorem in question is given. In the proof of Theorem 2.3 in [2], the statement "that distinct L's cannot belong to a common pure subgroup HL follows immediately from the fact that no two L's generate with G [p] the same subgroup of G" is false as it stands. One has to exercise a little more care in the selection of HL. For example, since the union [xa, Xa]aEA of all the L's (in Lemma 2.1 in [2]) is still linearly independent, there exists according to [1] a pure subgroup H of G supported by S such that HDL for each L. Hence one could choose HL=H for each L. However, all goes well if HL is chosen such that HLn [Xa, XTg aEA =L, as can easily be done. Similar remarks hold for Lemma 3.8 in [2]. A short alternate proof of Theorem 2.3 in [2] is given. In what follows let c be the cardinality of the continuum and Q be the first ordinal whose cardinality is c. The notation and terminology will be the same as that in [2].
- Published
- 1969
28. Observations on a paper by Rosenblum
- Author
-
S. Cater
- Subjects
Complex conjugate ,Applied Mathematics ,General Mathematics ,Hilbert space ,Uniform limit theorem ,Combinatorics ,symbols.namesake ,Operator (computer programming) ,Skew-Hermitian matrix ,Bounded function ,symbols ,Normal operator ,Complex number ,Mathematics - Abstract
M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.
- Published
- 1961
29. A note to the paper on integral bases by H. B. Mann
- Author
-
Virginia Hanly and Henry B. Mann
- Subjects
Applied Mathematics ,General Mathematics ,Calculus ,Mathematics - Published
- 1958
30. Notes on a paper by Sanov. II
- Author
-
Ruth Rebekka Struik
- Subjects
Combinatorics ,Generalization ,Applied Mathematics ,General Mathematics ,Lie ring ,Prime (order theory) ,Mathematics - Abstract
F(k) ={xkj x F}; F,=F; Fk= (Fk_l,F). Let (u, v, 0)=u, (u, v, 1)=(u, v), (u, v, n)=((u, v, n-1), v). Then Sanov [3] proved that (1. 1) (u, v, apa _1) p-a E F(p#)Fapa+?, /3 a = 1, 2, where p is a prime. In this paper, (1.1) is proved for the cases a= 1, 2; 3 arbitrary. A slight generalization of these results is also proved. Sanov's proof involved an investigation of ideals in a Lie Ring. In this paper, Hall's Collection Process will be used. The method also yields other formulas, e.g. (1.2) (u, v, p2 1)P' (E F2p2_pF(p) (1.3) (u, v, pa+l 1)p~l C F2pa+l paF(pl), a = 1 2, and can be used to produce numerous formulas of a similar nature. The author hopes that some of these formulas and/or the method may be of use in solving other group-theoretic problems. The author was unable to use the method to prove (1.1) for a =3. Note that for a=f = 1, (1. 1) becomes (1.4) (U, v, p 1) E F(p)Fv+1. (1.4) plays an important role in the theory of the Restricted Burnside Problem.
- Published
- 1961
31. Note on a paper of J. L. Palacios: 'A correction note on: ‘Generalized Hewitt-Savage theorems for strictly stationary processes’ [Proc. Amer. Math. Soc. 63 (1977), no. 2, 313–316; MR0501304 (58 #18695)] by Isaac' [ibid. 88 (1983), no. 1, 138–140; MR0691294 (86a:60054)]
- Author
-
Richard Isaac
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Calculus ,Mathematics - Abstract
J. L. Palacios claimed [2] that the author’s paper [1] contained errors. This note refutes those claims by showing that Palacios misunderstood [1] and adopted assumptions different from those of [1].
- Published
- 1987
32. A note on a paper of Palais
- Author
-
L. Jonker
- Subjects
Combinatorics ,Closed and exact differential forms ,Lift (mathematics) ,Tensor product ,Differential form ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,De Rham cohomology ,Exterior derivative ,Mathematics ,Tensor field - Abstract
If BP denotes the space of smooth alternating p-forms on the C' manifold M, we are interested in finding the spaces of R-linear maps from iPP to be that commute with the diffeomorphisms of M. For a compact manifold M these spaces were found by R. S. Palais. In this note we find them for noncompact M. Let V, W be the spaces of sections of two tensor bundles over a connected Coo manifold M of dimension n. In a paper [3] appearing in Trans. Amer. Math. Soc. 92 (1959), R. S. Palais studied a number of the spaces 4 (V, W) of the R-linear maps V->W natural with respect to the action on V and W of the group G of diffeomorphisms on M. We sketch the definition of this action: If g-G we define R,:F(TM) -4F(TM) to be the differential of g, and Rg:r(T*M)-->F(T*M) to be the dual of the differential of g-1. For the other spaces of tensor fields Rg is defined as the obvious tensor product of these two R-linear isomorphisms. A map c: V->W is then called natural if cRg =R,c for all g G. Palais was particularly interested in the case when V and W are the space IP of p-forms and the space bq of q-forms respectively, 0 < p, q
- Published
- 1971
33. Errata for two papers of Stitzinger
- Author
-
Ernest L. Stitzinger
- Subjects
Discrete mathematics ,Direct sum ,Applied Mathematics ,General Mathematics ,Mistake ,Point (geometry) ,Invariant (mathematics) ,Notation ,Mathematics - Abstract
There is a mistake in each of [3] and [4]. The purpose of this note is to correct the error in [3] and to salvage what can be saved in [4]. In each case the notation will be that of the paper under discussion. Professor Homer Bechtell has kindly informed me of an error in the proof of the Theorem of [3] which occurs in the case MG1c G. The theorem is true however by altering the proof at this point. Assume that Mc: MG1=M1lc:G. M1 is an invariant subgroup of G, and since G1c: Soc(G), G1 is a direct sum of minimal invariant subgroups of G. By Clifford's theorem (p. 70 of [1]), each minimal invariant subgroup of G is either M1-central or M1-hypereccentric, and hence
- Published
- 1972
34. A note on my paper: 'On symmetric matrices whose eigenvalues satisfy linear inequalities'
- Author
-
Fritz John
- Subjects
Combinatorics ,Discrete mathematics ,Linear inequality ,Applied Mathematics ,General Mathematics ,Regular polygon ,Convex set ,Symmetric matrix ,Elementary symmetric polynomial ,Matrix analysis ,Invariant (mathematics) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The first part of the theorem of the above paper states that if ois a closed convex set in real X1 ... Xn-space which is invariant under permutations of coordinates, and if C(o-) denotes the set of real symmetric nXn matrices whose eigenvalues X1, X. form the coordinates of points in a, then C(o) is convex. I am obliged to Professor R. T. Rockafellar for pointing out that this statement is essentially contained in a theorem of Chandler Davis.2 The statement also follows from an earlier result of V. B. Lidskii,3 which was not known to me at the time of publication.
- Published
- 1968
35. Remarks on a paper of Hobby and Wright
- Author
-
Paul Hill
- Subjects
Combinatorics ,Wright ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Frattini subgroup ,Term (logic) ,Notation ,Central series ,Hobby ,Mathematics - Abstract
C. Hobby and C. R. B. Wright [2] have just published the following Theorem A. However, their proof seems to contain an error.2 The notation of [2] is used except that G. is not reserved for the nth term of the lower central series of G: +/(G) denotes the Frattini subgroup of G; (G, H) means the group generated by the commutators g-lh-1gh where gEG, hEH; (A1, A2, * * *, A.+,) is defined inductively as ((A1, A2, . , A.), A.+,); HCG means that H is properly included in G.
- Published
- 1962
36. A remark on Neuwirth and Newman’s paper: 'Positive 𝐻^{1/2} functions are constants'
- Author
-
Shinji Yamashita
- Subjects
Combinatorics ,Lemma (mathematics) ,symbols.namesake ,Applied Mathematics ,General Mathematics ,Blaschke product ,symbols ,Function (mathematics) ,Absolute value (algebra) ,Boundary values ,Decomposition theorem ,Mathematical physics ,Mathematics - Abstract
PROOF. By a theorem of Rudin a function gEH' in U whose boundary values are real a.e. on I can be analytically continued to D [3, p. 59]. The lemma follows on applying Rudin's result to gi= (1/2) (fl+f2) and g2=(i/2) (fi-f2). PROOF OF THEOREM 1. By a well-known decomposition theorem [2, p. 87], f(z)=B(z)F2(Z), where B(z) is a Blaschke product and F(z) EH1. Since the boundary values of B (z) have absolute value one a.e. on K, we have a.e. on I, f(ei0)= |f(eio)I, or B(ei0)F2(ei0) = F2(ei0) |, and hence
- Published
- 1969
37. Remark on S. N. Roy’s paper 'A useful theorem in matrix theory'
- Author
-
Béla Sz.-Nagy
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,Mathematics - Published
- 1956
38. A note on the Hitchin-Thorpe inequality and Ricci flow on 4-manifolds
- Author
-
Yuguang Zhang and Zhenlei Zhang
- Subjects
Mathematics - Differential Geometry ,Hitchin–Thorpe inequality ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Short paper ,Ricci flow ,Type inequality ,Differential Geometry (math.DG) ,Bounded function ,FOS: Mathematics ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,53C20, 53C44 ,Yamabe invariant ,Mathematics ,Scalar curvature - Abstract
In this short paper, we prove a Hitchin-Thorpe type inequality for closed 4-manifolds with non-positive Yamabe invariant, and admitting long time solutions of the normalized Ricci flow equation with bounded scalar curvature.
- Published
- 2012
39. On the classification of solutions of -Δ𝑢=𝑒^{𝑢} on ℝ^{ℕ}: Stability outside a compact set and applications
- Author
-
Alberto Farina and E. N. Dancer
- Subjects
Pure mathematics ,Compact space ,Euclidean space ,Applied Mathematics ,General Mathematics ,Bounded function ,Short paper ,Topology ,Stability (probability) ,Domain (mathematical analysis) ,Mathematics - Abstract
In this short paper we prove that, for3≤N≤93 \le N \le 9, the problem−Δu=eu-\Delta u = e^uon the entire Euclidean spaceRN\mathbb {R}^Ndoes not admit any solution stable outside a compact set ofRN\mathbb {R}^N. This result is obtained without making any assumption about the boundedness of solutions. Furthermore, as a consequence of our analysis, we also prove the non-existence of finite Morse Index solutions for the considered problem. We then use our results to give some applications to bounded domain problems.
- Published
- 2008
40. A new proof of Mok’s generalized Frankel conjecture theorem
- Author
-
Hui-Ling Gu
- Subjects
Pure mathematics ,Conjecture ,Maximum principle ,Simple (abstract algebra) ,Applied Mathematics ,General Mathematics ,Short paper ,Calculus ,Maximal principle ,Mathematics::Differential Geometry ,Transcendental number ,Mathematics - Abstract
In this short paper, we will give a simple and transcendental proof for Mok's theorem of the generalized Frankel conjecture. This work is based on the maximum principle proposed by Brendle and Schoen.
- Published
- 2008
41. Integers represented as the sum of one prime, two squares of primes and powers of 2
- Author
-
Haiwei Sun and Guangshi Lü
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Short paper ,MathematicsofComputing_GENERAL ,Prime number ,Prime (order theory) ,Algebra ,symbols.namesake ,Integer ,symbols ,Idoneal number ,Prime power ,Sphenic number ,Mathematics - Abstract
In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 83 83 powers of 2 2 .
- Published
- 2008
42. Homological stability of non-orientable mapping class groups with marked points
- Author
-
Elizabeth Hanbury
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Algebraic Geometry ,Applied Mathematics ,General Mathematics ,Short paper ,Homology (mathematics) ,Mathematics::Geometric Topology ,Mapping class group ,Mathematics - Abstract
Wahl recently proved that the homology of the non-orientable mapping class group stabilizes as the genus increases. In this short paper we analyse the situation where the underlying non-orientable surfaces have marked points.
- Published
- 2008
43. On the degree two entry of a Gorenstein $h$-vector and a conjecture of Stanley
- Author
-
Fabrizio Zanello, Juan C. Migliore, and Uwe Nagel
- Subjects
Combinatorics ,Conjecture ,Integer ,Degree (graph theory) ,Applied Mathematics ,General Mathematics ,Short paper ,Codimension ,h-vector ,Upper and lower bounds ,Unimodality ,Mathematics - Abstract
In this short paper we establish a (non-trivial) lower bound on the degree two entry h 2 of a Gorenstein h-vector of any given socle degree e and any codimension r. In particular, when e = 4, that is, for Gorenstein h-vectors of the form h = (1, r, h 2 , r, 1), our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say f(r), that h 2 may assume. In fact, we show that limr→∞ f(r) r 2/3 =6 2/3 . In general, we wonder whether our lower bound is sharp for all integers e > 4 and r > 2.
- Published
- 2008
44. Higher order Turán inequalities for the Riemann $\xi$-function
- Author
-
Dimitar K. Dimitrov, Fábio Rodrigues Lucas, Universidade Estadual Paulista (Unesp), and Universidade Estadual de Campinas (UNICAMP)
- Subjects
Degree (graph theory) ,Applied Mathematics ,General Mathematics ,Entire function ,Mathematical analysis ,Short paper ,Function (mathematics) ,Maclaurin coefficients ,Riemann ξ function ,Combinatorics ,Riemann hypothesis ,symbols.namesake ,Jensen polynomials ,symbols ,Order (group theory) ,Shape function ,Laguerre-Pólya class ,Turán inequalities ,Mathematics - Abstract
Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:25:28Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:41:41Z : No. of bitstreams: 1 2-s2.0-79951846250.pdf: 494002 bytes, checksum: 56b6ee8beddda3e7dae971355d44a19f (MD5) Made available in DSpace on 2014-05-27T11:25:28Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-03-01 Item merged in doublecheck by Felipe Arakaki (arakaki@reitoria.unesp.br) on 2015-12-11T17:28:11Z Item was identical to item(s): 71803, 21370 at handle(s): http://hdl.handle.net/11449/72321, http://hdl.handle.net/11449/21804 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) The simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society. Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP Departamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SP Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP FAPESP: 03/01874-2 FAPESP: 06/60420-0 CNPq: 305622/2009-9 CAPES: DGU-160
- Published
- 2011
45. The Gottlieb group of finite linear quotients of odd-dimensional spheres
- Author
-
S. Allen Broughton
- Subjects
Pure mathematics ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Homotopy ,Short paper ,Geometry ,SPHERES ,Isomorphism ,Geometric proof ,Quotient ,Homeomorphism ,Mathematics - Abstract
Let G be a finite, freely acting group of homeomorphisms of the odd-dimensional sphere S 2n−1 . John Oprea has proven that the Gottlieb group of S 2n−1 /G equals Z(G), the centre of G. The purpose of this short paper is to give a considerably shorter, more geometric proof of Oprea's theorem in the important case where G is a linear group
- Published
- 1991
46. On the first factor of the class number of a cyclotomic field
- Author
-
Ke Qin Feng
- Subjects
Combinatorics ,Stickelberger's theorem ,Applied Mathematics ,General Mathematics ,Short paper ,Herbrand–Ribet theorem ,Order (group theory) ,Field (mathematics) ,Cyclotomic field ,Class number ,Prime (order theory) ,Mathematics - Abstract
Let p p be an odd prime. h 1 ( p ) {h_1}(p) is the first factor of the class number of field Q ( ζ p ) Q({\zeta _p}) . We proved that \[ h 1 ( p ) ⩽ { 2 p ( p − 1 8 ( 2 l / 2 + 1 ) 4 / l ) ( p − 1 ) / 4 , if l is even, 2 p ( p − 1 8 ( 2 l − 1 ) 2 / l ) ( p − 1 ) / 4 , if l is odd . {h_1}(p) \leqslant \left \{ \begin {gathered} 2p{\left ( {\frac {{p - 1}} {{8{{({2^{l/2}} + 1)}^{4/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is even,}} \hfill \\ 2p{\left ( {\frac {{p - 1}} {{8{{({2^l} - 1)}^{2/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is odd}}{\text {.}} \hfill \\ \end {gathered} \right . \] From that we obtain h 1 ( p ) ⩽ 2 p ( ( p − 1 ) / 31.997158 … ) ( p − 1 ) / 4 {h_1}(p) \leqslant 2p{((p - 1)/31.997158 \ldots )^{(p - 1)/4}} which is better than Carlitz’s and Metsänkyla’s results. For the fields Q ( ζ 2 n ) Q({\zeta _{{2^n}}}) and Q ( ζ p n ) ( n ⩾ 2 ) Q({\zeta _{{p^n}}})(n \geqslant 2) , we get the similar results.
- Published
- 1982
47. Covering by homothets and illuminating convex bodies
- Author
-
Alexey Glazyrin
- Subjects
Conjecture ,Applied Mathematics ,General Mathematics ,Discrete geometry ,Boundary (topology) ,Metric Geometry (math.MG) ,Upper and lower bounds ,Infimum and supremum ,Homothetic transformation ,Combinatorics ,Mathematics - Metric Geometry ,Hausdorff dimension ,FOS: Mathematics ,Mathematics::Metric Geometry ,Convex body ,Mathematics - Abstract
The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $\alpha$ and a convex body $B$, $g_{\alpha}(B)$ is the infimum of $\alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{\alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $\alpha$. In this paper, we prove that $g_{\alpha}(B)\leq h_{\alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{\alpha} (B) > 2^{d-\alpha}$ for almost all $\alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.
- Published
- 2021
48. On the Baum–Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg conjecture
- Author
-
Adam Skalski and Yuki Arano
- Subjects
Pure mathematics ,Conjecture ,Mathematics::Operator Algebras ,Quantum group ,Applied Mathematics ,General Mathematics ,Mathematics - Operator Algebras ,Crossed product ,Unimodular matrix ,Mathematics::K-Theory and Homology ,Primary 46L67, Secondary 46L80 ,FOS: Mathematics ,Baum–Connes conjecture ,Countable set ,Equivariant map ,Operator Algebras (math.OA) ,Quantum ,Mathematics - Abstract
We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group $\Gamma$ implies that $\Gamma$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition., Comment: 15 pages, v2 corrects a few minor points. The final version of the paper will appear in the Proceedings of the American Mathematical Society
- Published
- 2021
49. The nilpotent cone for classical Lie superalgebras
- Author
-
Daniel K. Nakano and L. Jenkins
- Subjects
Pure mathematics ,Nilpotent cone ,17B20, 17B10 ,Applied Mathematics ,General Mathematics ,Group Theory (math.GR) ,Representation theory ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper the authors introduce an analogue of the nilpotent cone, N {\mathcal N} , for a classical Lie superalgebra, g {\mathfrak g} , that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, g = g 0 ¯ ⊕ g 1 ¯ {\mathfrak g}={\mathfrak g}_{\bar 0}\oplus {\mathfrak g}_{\bar 1} with Lie G 0 ¯ = g 0 ¯ \text {Lie }G_{\bar 0}={\mathfrak g}_{\bar 0} , it is shown that there are finitely many G 0 ¯ G_{\bar 0} -orbits on N {\mathcal N} . Later the authors prove that the Duflo-Serganova commuting variety, X {\mathcal X} , is contained in N {\mathcal N} for any classical simple Lie superalgebra. Consequently, our finiteness result generalizes and extends the work of Duflo-Serganova on the commuting variety. Further applications are given at the end of the paper.
- Published
- 2021
50. Corrigenda to 'Cohen-Macaulay bipartite graphs in arbitrary codimension'
- Author
-
Rahim Zaare-Nahandi, Hassan Haghighi, and Siamak Yassemi
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,Bipartite graph ,Codimension ,Mathematics - Abstract
A misuse of terminology has occurred in the statement and proof of Theorem 4.1 in our paper [Proc. Amer. Math. Soc. 143 (2015), pp. 1981–1989] which caused some justifiable misinterpretation of this result. To recover this result we provide a new definition and give the statement of our result in terms of this definition. The proof of the new version is an improvement of the old proof. The effect of the new definition on further relevant results in our paper has been adopted in a remark.
- Published
- 2021
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