What Is the Complexity of Surface Integration?

We study the worst case complexity of computing ε-approximations of surface integrals. This problem has two sources of partial information: the integrand f and the function g defining the surface. The problem is nonlinear in its dependence on g. Here, f is an r times continuously differentiable scalar function of l variables, and g is an s times continuously differentiable injective function of d variables with l components. We must have d⩽l and s⩾1 for surface integration to be well-defined. Surface integration is related to the classical integration problem for functions of d variables that are min{r, s−1} times continuously differentiable. This might suggest that the complexity of surface integration should be Θ((1/ε)^{d/min{r, s−1}}). Indeed, this holds when d<l and s=1, in which case the surface integration problem has infinite complexity. However, if d⩽l and s⩾2, we prove that the complexity of surface integration is O((1/ε)^d/min{r, s}). Furthermore, this bound is sharp whenever d<l.     

Small maximally disjoint union-free families

A family F of k-subsets of an n-set X is disjoint union-free (DUF) if all disjoint pairs of elements of F have distinct unions; that is, if for every A,B,C,D∈F, A∩B=C∩D=∅ and A∪B=C∪D implies {A,B}={C,D}. DUF families of maximum size have been studied by Erdös and Füredi. Let F be DUF with the property that F∪{E} is not DUF for any k-subset E of X not already in F. Then F is maximally DUF. We introduce the problem of finding the minimum size of maximally DUF families and provide bounds on this quantity for k=3.     

Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps

Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbS^d\mathbb{S}^{d} of ℝ ^d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbS^dx,y\in\mathbb{S}^{d} . Given a spherical cap

B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      4. Resolvable Mendelsohn triple systems with equal sized holes      F. E. Bennett  R. Wei  L. Zhu 《组合设计杂志》1997,5(5):329-340 An HMTS of type {n1, n2, ⋖, nh} is a directed graph which can be decomposed into 3-circuits. If the 3-circuits can be partitioned into parallel classes, then the HMTS is called an RHMTS. In this article it is shown that the RHMTSs of type mh exist when mh &equiv 0 (mod 3) and (m, h) &ne (1, 6), with the possible exception of h = 6 and , where M17 = {m|m is divisible by a prime less than 17}. The existence of Mendelsohn frames, which is closely related to RHMTS, is also considered in this article. It is proved that a Mendelsohn frame of type tu exists if and only if u ≥ 4 and t(u - 1) ≡ 0(mod 3) with 2 possible exceptions. © 1997 John Wiley & Sons, Inc. J Combin Designs 5:329–340, 1997      5. Rose window graphs underlying rotary maps      István Kovács  Klavdija Kutnar 《Discrete Mathematics》2010,310(12):1802-1811 Given natural numbers n≥3 and 1≤a,r≤n−1, the rose window graph Rn(a,r) is a quartic graph with vertex set {xi∣i∈Zn}∪{yi∣i∈Zn} and edge set {{xi,xi+1}∣i∈Zn}∪{{yi,yi+r}∣i∈Zn}∪{{xi,yi}∣i∈Zn}∪{{xi+a,yi}∣i∈Zn}. In this paper rotary maps on rose window graphs are considered. In particular, we answer the question posed in [S. Wilson, Rose window graphs, Ars Math. Contemp. 1 (2008), 7-19. http://amc.imfm.si/index.php/amc/issue/view/5] concerning which of these graphs underlie a rotary map.      6. Quasi-convex sequences in the circle and the 3-adic integers      D. Dikranjan  Gábor Lukács 《Topology and its Applications》2010,157(8):1357-1369 In this paper, we present families of quasi-convex sequences converging to zero in the circle group T, and the group J3 of 3-adic integers. These sequences are determined by increasing sequences of integers. For an increasing sequence , put gn=an+1−an. We prove that (a) the set {0}∪{±3−(an+1)|n∈N} is quasi-convex in T if and only if a0>0 and gn>1 for every n∈N; (b) the set {0}∪{±an3|n∈N} is quasi-convex in the group J3 of 3-adic integers if and only if gn>1 for every n∈N. Moreover, we solve an open problem from [D. Dikranjan, L. de Leo, Countably infinite quasi-convex sets in some locally compact abelian groups, Topology Appl. 157 (8) (2010) 1347-1356] providing a complete characterization of the sequences such that {0}∪{±2−(an+1)|n∈N} is quasi-convex in T. Using this result, we also obtain a characterization of the sequences such that the set {0}∪{±2−(an+1)|n∈N} is quasi-convex in R.      7. 可数簇全拟-Φ-渐近非扩张映象和广义混合平衡问题以及极大单调算子的收缩投影迭代算法          下载免费PDF全文 朱浸华 《数学物理学报(A辑)》2014,34(2):283-302 先介绍全拟-φ-渐近非扩张映象的概念,然后在具有Kadec—Klee性质的一致光滑、严格凸的Banach空间的框架下,利用混合收缩投影的迭代算法,用以寻求广义混合平衡问题的解集GMEP,可数簇全拟-φ-渐近非扩张映象的不动点集(?)F(S_(i))和极大单调算子的零点集T~(-1)0的公共元.在适当的条件下,证明了逼近于这一公共元的强收敛定理.推广和改进了一些最新结果.      8. Definability in the Substructure Ordering of Simple Graphs      Alexander Wires 《Annals of Combinatorics》2016,20(1):139-176 For simple graphs, we investigate and seek to characterize the properties first-order definable by the induced subgraph relation. Let \({\mathcal{P}\mathcal{G}}\) denote the set of finite isomorphism types of simple graphs ordered by the induced subgraph relation. We prove this poset has only one non-identity automorphism co, and for each finite isomorphism type G, the set {G, G co } is definable. Furthermore, we show first-order definability in \({\mathcal{P}\mathcal{G}}\) captures, up to isomorphism, full second-order satisfiability among finite simple graphs. These results can be utilized to explore first-order definability in the closely associated lattice of universal classes. We show that for simple graphs, the lattice of universal classes has only one non-trivial automorphism, the set of finitely generated and finitely axiomatizable universal classes are separately definable, and each such universal subclass is definable up to the unique non-trivial automorphism.      9. Some results on non-linear recurrence      V. Bergelson  M. Boshernitzan  J. Bourgain 《Journal d'Analyse Mathématique》1994,62(1):29-46 We study the behavior of measure-preserving systems with continuous time along sequences of the form {n α}n∈#x2115;} where α is a positive real number1. Let {S t } t∈? be an ergodic continuous measure preserving flow on a probability Lebesgue space (X, β, μ). Among other results we show that: For all but countably many α (in particular, for all α∈???) one can find anL ∞-functionf for which the averagesA N (f)(1/N)=Σ n=1 N f(S nα x) fail to converge almost everywhere (the convergence in norm holds for any α!). For any non-integer and pairwise distinct numbers α1, α2,..., α k ∈(0, 1) and anyL ∞-functionsf 1,f 2, ...,f k , one has $$\mathop {lim}\limits_{N \to \infty } \left\| {\frac{1}{N}\sum\limits_{n - 1}^N {\prod\limits_{i - 1}^k {f_i (S^{n^{\alpha _i } } x) - \prod\limits_{i - 1}^k {\int_X {f_i d\mu } } } } } \right\|_{L^2 } = 0$$ We also show that Furstenberg’s correspondence principle fails for ?-actions by demonstrating that for all but a countably many α>0 there exists a setE?? having densityd(E)=1/2 such that, for alln∈?, $$d(E \cap (E - n^\alpha )) = 0$$ .      10. Hamilton cycles in digraphs of unitary matrices      G. Gutin  A. Yeo 《Discrete Mathematics》2006,306(24):3315-3320 A set S⊆V is called a q+-set (q--set, respectively) if S has at least two vertices and, for every u∈S, there exists v∈S,v≠u such that N+(u)∩N+(v)≠∅ (N-(u)∩N-(v)≠∅, respectively). A digraph D is called s-quadrangular if, for every q+-set S, we have |∪{N+(u)∩N+(v):u≠v,u,v∈S}|?|S| and, for every q--set S, we have |∪{N-(u)∩N-(v):u,v∈S)}?|S|. We conjecture that every strong s-quadrangular digraph has a Hamilton cycle and provide some support for this conjecture.      11. A note on an inverse problem for lattice points      ?eljka Ljuji? Camilo Sanabria 《Topology and its Applications》2011,158(8):1012-1018 Let K⊆R2 be a compact set such that K+Z2=R2 and let K−K={a−b|a,b∈K}. We prove, via Algebraic Topology, that the integer points of the difference set of K, (K−K)∩Z2, is not contained on the coordinate axes, Z×{0}∪{0}×Z. This result implies the negative answer to the inverse problem posed by M.B. Nathanson (2010) [5].      12. Some cyclic codes of length 2<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>      Sudhir Batra  S. K. Arora 《Designs, Codes and Cryptography》2011,61(1):41-69 Explicit expressions for 4n + 2 primitive idempotents in the semi-simple group ring $R_{2p^{n}}\equiv \frac{GF(q)[x]}{p and q are distinct odd primes; n ≥ 1 is an integer and q has order \fracf(2pn)2{\frac{\phi(2p^{n})}{2}} modulo 2p n . The generator polynomials, the dimension, the minimum distance of the minimal cyclic codes of length 2p n generated by these 4n + 2 primitive idempotents are discussed. For n = 1, the properties of some (2p, p) cyclic codes, containing the above minimal cyclic codes are analyzed in particular. The minimum weight of some subset of each of these (2p, p) codes are observed to satisfy a square root bound.      13. Discrepancy of Arithmetic Progressions in Higher Dimensions      Benedek Valkó 《Journal of Number Theory》2002,92(1):117-130 K. F. Roth (1964, Acta. Arith.9, 257-260) proved that the discrepancy of arithmetic progressions contained in [1, N]={1, 2, …, N} is at least cN1/4, and later it was proved that this result is sharp. We consider the d-dimensional version of this problem. We give a lower estimate for the discrepancy of arithmetic progressions on [1, N]d and prove that this result is nearly sharp. We use our results to give an upper estimate for the discrepancy of lines on an N×N lattice, and we also give an estimate for the discrepancy of a related random hypergraph.      14. Eigenvalue comparisons for second order difference equations with periodic and antiperiodic boundary conditions      Jun Ji  Bo Yang 《Journal of Applied Mathematics and Computing》2008,27(1-2):307-324 We consider a class of boundary value problems of the second order difference equation $$\Delta(r_{i-1}\Delta y_{i-1})-b_{i}y_{i}+\lambda a_{i}y_{i}=0,\quad 1\le i\le n,\ y_{0}=\alpha y_{n},\ y_{n+1}=\alpha y_{1}.$$ The class of problems considered includes those with antiperiodic, Dirichlet, and periodic boundary conditions. We focus on the structure of eigenvalues of this class of problems and comparisons of all eigenvalues as the coefficients {a i } i=1 n ,{b i } i=1 n , and {r i } i=0 n change.      15. Uniform approximation on compact subsets of the real line      M. B. Korobkova 《Mathematical Notes》1974,16(2):779-785 Let {? i } i=∩ n be continuous real functions on the compact set M?R. We consider the problem of best uniform approximation of the function? by polynomials \(\sum\nolimits_{i = 1}^n {c_i \varphi _i }\) on M. Let V(?0, A) be a set of polynomials of best approximation on A ? M. We show that \(V(\varphi _0 ,M) = \mathop \cap \limits_{A_{n + 1} } V(\varphi _0 ,A_{n + 1} )\) , where An+1 represents all the possible sets of n+ 1 points {x1, ..., xn+1} in M, containing the characteristic set of the given problem of best approximation and for which the the rank of ∥?i ∥ (i=1, ...,n; j=1,..., n+1) is equal to n. This theorem is applied to a problem of uniform approximation where {? i } i=1 n is a weakly Chebyshev system.      16. A product of varieties in relation to Baer invariants      Mohsen Parvizi 《Indagationes Mathematicae》2006,17(2):297-309 The concept of the Baer invariant is useful in classifying groups into isologism classes. In this paper two sequences of varieties {rVn}n∈N0 and {lVn}n∈N0, are considered from a given variety V. The structure of Baer invariants of some groups with respect to these varieties, are determined for some specific V.      17. Cross-intersecting families and primitivity of symmetric systems      Jun Wang 《Journal of Combinatorial Theory, Series A》2011,118(2):455-462       18. Where Does Smoothness Count the Most for Two-Point Boundary-Value Problems?      《Journal of Complexity》1999,15(3):360-384 We study the complexity of scalar 2mth order elliptic two-point boundary-value problems Lu=f, error being measured in the energy norm. Previous work on the complexity of these problems has generally assumed that we had partial information about the right-hand side f and complete information about the coefficients of L. In this paper, we study the complexity of such problems when, in addition to partial information about f, we have only partial information about the coefficients of L. More precisely, we suppose that f has r derivatives in the Lp-sense, with r⩾−m and p∈[2, ∞], and that L has the usual divergence form Lv=∑0⩽i, j⩽m (−1)i Di(ai, j Djv), with ai, j being ri, j-times continuously differentiable, where ri, j⩾0. We first suppose that continuous linear information is available. Let r=min{r, min0⩽i, j⩽m {ri, j−i}}. If r=−m, the problem is unsolvable; for r>−m, we find that the ε-complexity is proportional to (1/ε)1/(r+m), and we show that a finite element method (FEM) is optimal. We next suppose that only standard information (consisting of function and/or derivative evaluations) is available. Let rmin=min{r, min0⩽i, j⩽m {ri, j}}. If rmin=0, the problem is unsolvable; for rmin>0, we find that the ε-complexity is proportional to (1/ε)1/rmin, and we show that a modified FEM (which uses only function evaluations, and not derivatives) is optimal.      19. There are no <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-stable intersections of regular Cantor sets      Carlos Gustavo Moreira 《Acta Mathematica》2011,206(2):311-323       20. Variations of Y-dominating functions on graphs      Chuan-Min Lee 《Discrete Mathematics》2008,308(18):4185-4204 Let Y be a subset of real numbers. A Y-dominating function of a graph G=(V,E) is a function f:V→Y such that for all vertices v∈V, where NG[v]={v}∪{u|(u,v)∈E}. Let for any subset S of V and let f(V) be the weight of f. The Y-domination problem is to find a Y-dominating function of minimum weight for a graph G=(V,E). In this paper, we study the variations of Y-domination such as {k}-domination, k-tuple domination, signed domination, and minus domination for some classes of graphs. We give formulas to compute the {k}-domination, k-tuple domination, signed domination, and minus domination numbers of paths, cycles, n-fans, n-wheels, n-pans, and n-suns. Besides, we present a unified approach to these four problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. This paper also gives complexity results for the problems on doubly chordal graphs, dually chordal graphs, bipartite planar graphs, chordal bipartite graphs, and planar graphs.

Resolvable Mendelsohn triple systems with equal sized holes

An HMTS of type {n₁, n₂, ⋖, n_h} is a directed graph which can be decomposed into 3-circuits. If the 3-circuits can be partitioned into parallel classes, then the HMTS is called an RHMTS. In this article it is shown that the RHMTSs of type m^h exist when mh &equiv 0 (mod 3) and (m, h) &ne (1, 6), with the possible exception of h = 6 and , where M₁₇ = {m|m is divisible by a prime less than 17}. The existence of Mendelsohn frames, which is closely related to RHMTS, is also considered in this article. It is proved that a Mendelsohn frame of type t^u exists if and only if u ≥ 4 and t(u - 1) ≡ 0(mod 3) with 2 possible exceptions. © 1997 John Wiley & Sons, Inc. J Combin Designs 5:329–340, 1997     

Rose window graphs underlying rotary maps

Given natural numbers n≥3 and 1≤a,r≤n−1, the rose window graph R_n(a,r) is a quartic graph with vertex set {x_i∣i∈Z_n}∪{y_i∣i∈Z_n} and edge set {{x_i,x_i+1}∣i∈Z_n}∪{{y_i,y_i+r}∣i∈Z_n}∪{{x_i,y_i}∣i∈Z_n}∪{{x_i+a,y_i}∣i∈Z_n}. In this paper rotary maps on rose window graphs are considered. In particular, we answer the question posed in [S. Wilson, Rose window graphs, Ars Math. Contemp. 1 (2008), 7-19. http://amc.imfm.si/index.php/amc/issue/view/5] concerning which of these graphs underlie a rotary map.     

Quasi-convex sequences in the circle and the 3-adic integers

In this paper, we present families of quasi-convex sequences converging to zero in the circle group T, and the group J₃ of 3-adic integers. These sequences are determined by increasing sequences of integers. For an increasing sequence , put gn=an+1−an. We prove that

(a)

the set {0}∪{±³−(an+1)|n∈N} is quasi-convex in T if and only if a₀>0 and gn>1 for every n∈N;

(b)

the set {0}∪{±an³|n∈N} is quasi-convex in the group J₃ of 3-adic integers if and only if gn>1 for every n∈N.

Moreover, we solve an open problem from [D. Dikranjan, L. de Leo, Countably infinite quasi-convex sets in some locally compact abelian groups, Topology Appl. 157 (8) (2010) 1347-1356] providing a complete characterization of the sequences such that {0}∪{±²−(an+1)|n∈N} is quasi-convex in T. Using this result, we also obtain a characterization of the sequences such that the set {0}∪{±²−(an+1)|n∈N} is quasi-convex in R.     

可数簇全拟-Φ-渐近非扩张映象和广义混合平衡问题以及极大单调算子的收缩投影迭代算法

先介绍全拟-φ-渐近非扩张映象的概念,然后在具有Kadec—Klee性质的一致光滑、严格凸的Banach空间的框架下,利用混合收缩投影的迭代算法,用以寻求广义混合平衡问题的解集GMEP,可数簇全拟-φ-渐近非扩张映象的不动点集(?)F(S_(i))和极大单调算子的零点集T~(-1)0的公共元.在适当的条件下,证明了逼近于这一公共元的强收敛定理.推广和改进了一些最新结果.     

Definability in the Substructure Ordering of Simple Graphs

For simple graphs, we investigate and seek to characterize the properties first-order definable by the induced subgraph relation. Let \({\mathcal{P}\mathcal{G}}\) denote the set of finite isomorphism types of simple graphs ordered by the induced subgraph relation. We prove this poset has only one non-identity automorphism co, and for each finite isomorphism type G, the set {G, G ^co } is definable. Furthermore, we show first-order definability in \({\mathcal{P}\mathcal{G}}\) captures, up to isomorphism, full second-order satisfiability among finite simple graphs. These results can be utilized to explore first-order definability in the closely associated lattice of universal classes. We show that for simple graphs, the lattice of universal classes has only one non-trivial automorphism, the set of finitely generated and finitely axiomatizable universal classes are separately definable, and each such universal subclass is definable up to the unique non-trivial automorphism.     

Some results on non-linear recurrence

We study the behavior of measure-preserving systems with continuous time along sequences of the form {n ^α}n∈#x2115;} where α is a positive real number¹. Let {S ^t } _t∈? be an ergodic continuous measure preserving flow on a probability Lebesgue space (X, β, μ). Among other results we show that:

1. For all but countably many α (in particular, for all α∈???) one can find anL ^∞-functionf for which the averagesA _N (f)(1/N)=Σ _n=1 ^N f(S ^nα x) fail to converge almost everywhere (the convergence in norm holds for any α!).
2. For any non-integer and pairwise distinct numbers α₁, α₂,..., α _k ∈(0, 1) and anyL ^∞-functionsf ₁,f ₂, ...,f _k , one has $$\mathop {lim}\limits_{N \to \infty } \left\| {\frac{1}{N}\sum\limits_{n - 1}^N {\prod\limits_{i - 1}^k {f_i (S^{n^{\alpha _i } } x) - \prod\limits_{i - 1}^k {\int_X {f_i d\mu } } } } } \right\|_{L^2 } = 0$$

We also show that Furstenberg’s correspondence principle fails for ?-actions by demonstrating that for all but a countably many α>0 there exists a setE?? having densityd(E)=1/2 such that, for alln∈?, $$d(E \cap (E - n^\alpha )) = 0$$ .     

Hamilton cycles in digraphs of unitary matrices

A set S⊆V is called a q⁺-set (q^--set, respectively) if S has at least two vertices and, for every u∈S, there exists v∈S,v≠u such that N⁺(u)∩N⁺(v)≠∅ (N^-(u)∩N^-(v)≠∅, respectively). A digraph D is called s-quadrangular if, for every q⁺-set S, we have |∪{N⁺(u)∩N⁺(v):u≠v,u,v∈S}|?|S| and, for every q^--set S, we have |∪{N^-(u)∩N^-(v):u,v∈S)}?|S|. We conjecture that every strong s-quadrangular digraph has a Hamilton cycle and provide some support for this conjecture.     

A note on an inverse problem for lattice points

Let K⊆R² be a compact set such that K+Z²=R² and let K−K={a−b|a,b∈K}. We prove, via Algebraic Topology, that the integer points of the difference set of K, (K−K)∩Z², is not contained on the coordinate axes, Z×{0}∪{0}×Z. This result implies the negative answer to the inverse problem posed by M.B. Nathanson (2010) [5].     

Some cyclic codes of length 2<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Explicit expressions for 4n + 2 primitive idempotents in the semi-simple group ring $R_{2p^{n}}\equiv \frac{GF(q)[x]}{p and q are distinct odd primes; n ≥ 1 is an integer and q has order \fracf(2pⁿ)2{\frac{\phi(2p^{n})}{2}} modulo 2p ⁿ . The generator polynomials, the dimension, the minimum distance of the minimal cyclic codes of length 2p ⁿ generated by these 4n + 2 primitive idempotents are discussed. For n = 1, the properties of some (2p, p) cyclic codes, containing the above minimal cyclic codes are analyzed in particular. The minimum weight of some subset of each of these (2p, p) codes are observed to satisfy a square root bound.     

Discrepancy of Arithmetic Progressions in Higher Dimensions

K. F. Roth (1964, Acta. Arith.9, 257-260) proved that the discrepancy of arithmetic progressions contained in [1, N]={1, 2, …, N} is at least cN^1/4, and later it was proved that this result is sharp. We consider the d-dimensional version of this problem. We give a lower estimate for the discrepancy of arithmetic progressions on [1, N]^d and prove that this result is nearly sharp. We use our results to give an upper estimate for the discrepancy of lines on an N×N lattice, and we also give an estimate for the discrepancy of a related random hypergraph.     

Eigenvalue comparisons for second order difference equations with periodic and antiperiodic boundary conditions

We consider a class of boundary value problems of the second order difference equation $$\Delta(r_{i-1}\Delta y_{i-1})-b_{i}y_{i}+\lambda a_{i}y_{i}=0,\quad 1\le i\le n,\ y_{0}=\alpha y_{n},\ y_{n+1}=\alpha y_{1}.$$ The class of problems considered includes those with antiperiodic, Dirichlet, and periodic boundary conditions. We focus on the structure of eigenvalues of this class of problems and comparisons of all eigenvalues as the coefficients {a _i } _i=1 ⁿ ,{b _i } _i=1 ⁿ , and {r _i } _i=0 ⁿ change.     

Uniform approximation on compact subsets of the real line

Let {? _i } _i=∩ ⁿ be continuous real functions on the compact set M?R. We consider the problem of best uniform approximation of the function? by polynomials \(\sum\nolimits_{i = 1}^n {c_i \varphi _i }\) on M. Let V(?₀, A) be a set of polynomials of best approximation on A ? M. We show that \(V(\varphi _0 ,M) = \mathop \cap \limits_{A_{n + 1} } V(\varphi _0 ,A_{n + 1} )\) , where A_n+1 represents all the possible sets of n+ 1 points {x₁, ..., x_n+1} in M, containing the characteristic set of the given problem of best approximation and for which the the rank of ∥?_i ∥ (i=1, ...,n; j=1,..., n+1) is equal to n. This theorem is applied to a problem of uniform approximation where {? _i } _i=1 ⁿ is a weakly Chebyshev system.     

A product of varieties in relation to Baer invariants

The concept of the Baer invariant is useful in classifying groups into isologism classes. In this paper two sequences of varieties {^rV_n}n∈N₀ and {^lV_n}n∈N₀, are considered from a given variety V. The structure of Baer invariants of some groups with respect to these varieties, are determined for some specific V.     

Where Does Smoothness Count the Most for Two-Point Boundary-Value Problems?

We study the complexity of scalar 2mth order elliptic two-point boundary-value problems Lu=f, error being measured in the energy norm. Previous work on the complexity of these problems has generally assumed that we had partial information about the right-hand side f and complete information about the coefficients of L. In this paper, we study the complexity of such problems when, in addition to partial information about f, we have only partial information about the coefficients of L. More precisely, we suppose that f has r derivatives in the L_p-sense, with r⩾−m and p∈[2, ∞], and that L has the usual divergence form Lv=∑_{0⩽i, j⩽m} (−1)ⁱ Dⁱ(a_i, j D^jv), with a_i, j being r_i, j-times continuously differentiable, where r_i, j⩾0. We first suppose that continuous linear information is available. Let r=min{r, min_{0⩽i, j⩽m} {r_i, j−i}}. If r=−m, the problem is unsolvable; for r>−m, we find that the ε-complexity is proportional to (1/ε)^1/(r+m), and we show that a finite element method (FEM) is optimal. We next suppose that only standard information (consisting of function and/or derivative evaluations) is available. Let r_min=min{r, min_{0⩽i, j⩽m} {r_i, j}}. If r_min=0, the problem is unsolvable; for r_min>0, we find that the ε-complexity is proportional to (1/ε)^1/r_min, and we show that a modified FEM (which uses only function evaluations, and not derivatives) is optimal.     

Variations of Y-dominating functions on graphs

Let Y be a subset of real numbers. A Y-dominating function of a graph G=(V,E) is a function f:V→Y such that for all vertices v∈V, where N_G[v]={v}∪{u|(u,v)∈E}. Let for any subset S of V and let f(V) be the weight of f. The Y-domination problem is to find a Y-dominating function of minimum weight for a graph G=(V,E). In this paper, we study the variations of Y-domination such as {k}-domination, k-tuple domination, signed domination, and minus domination for some classes of graphs. We give formulas to compute the {k}-domination, k-tuple domination, signed domination, and minus domination numbers of paths, cycles, n-fans, n-wheels, n-pans, and n-suns. Besides, we present a unified approach to these four problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. This paper also gives complexity results for the problems on doubly chordal graphs, dually chordal graphs, bipartite planar graphs, chordal bipartite graphs, and planar graphs.