17.—An Upper Bound for the Largest Zero of Hermite's Function with Applications to Subharmonic Functions

1976 ◽  
Vol 75 (3) ◽  
pp. 183-197 ◽  
Author(s):  
W. K Hayman ◽  
E. L Ortiz
Keyword(s):  
Upper Bound ◽  
Real Zero ◽  
Subharmonic Functions ◽  
Image Position

SYNOPSISLetbe Hermite's function of order λ and let h = h(λ) be the largest real zero of Hλ(t). SetIn this paper we establish the inequalitywhereEquality holds for S = ½. The result is also fairly accurate as S→0 and S→1. The proof is analytical except in the ranges −1·1 ≦ h ≦ −0·1 and where the argument is concluded by means of a computer.The following deduction is made elsewhere [2, Theorem A]. If u(x) is subharmonic in Rm(m ≧ 2) and the set E where u(x) > 0 has at least k components, where k ≧ 2, then the order ρ of u(x) is at least ϕ(1/k). In particular, if ρ < 1, E is connected. This result fails for ρ = 1.

Supremum and infimum of subharmonic functions of order between 1 and 2

2011 ◽  
Vol 54 (3) ◽  
pp. 685-693
Author(s):  
P. C. Fenton
Keyword(s):  
Lower Bound ◽  
Convex Function ◽  
Upper Bound ◽  
Counting Function ◽  
Subharmonic Functions ◽  
Image Position ◽  
Sharp Lower Bound

AbstractFor functions u, subharmonic in the plane, letand let N(r,u) be the integrated counting function. Suppose that $\mathcal{N}\colon[0,\infty)\rightarrow\mathbb{R}$ is a non-negative non-decreasing convex function of log r for which $\mathcal{N}(r)=0$ for all small r and $\limsup_{r\to\infty}\log\mathcal{N}(r)/\4\log r=\rho$, where 1 < ρ < 2, and defineA sharp upper bound is obtained for $\liminf_{r\to\infty}\mathcal{B}(r,\mathcal{N})/\mathcal{N}(r)$ and a sharp lower bound is obtained for $\limsup_{r\to\infty}\mathcal{A}(r,\mathcal{N})/\mathcal{N}(r)$.

Isolated minima of the product of n linear forms

1953 ◽  
Vol 49 (1) ◽  
pp. 59-62 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Forms ◽  
Image Position

Letbe n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.

Post completeness in modal logic

1972 ◽  
Vol 37 (4) ◽  
pp. 711-715 ◽  
Author(s):  
Krister Segerberg
Keyword(s):  
Modal Logic ◽  
Upper Bound ◽  
Modus Ponens ◽  
Proper Subset ◽  
Proper Extension ◽  
Image Position

Let ⊥, →, and □ be primitive, and let us have a countable supply of propositional letters. By a (modal) logic we understand a proper subset of the set of all formulas containing every tautology and being closed under modus ponens and substitution. A logic is regular if it contains every instance of □A ∧ □B ↔ □(A ∧ B) and is closed under the ruleA regular logic is normal if it contains □⊤. The smallest regular logic we denote by C (the same as Lemmon's C2), the smallest normal one by K. If L and L' are logics and L ⊆ L′, then L is a sublogic of L', and L' is an extension of L; properly so if L ≠ L'. A logic is quasi-regular (respectively, quasi-normal) if it is an extension of C (respectively, K).A logic is Post complete if it has no proper extension. The Post number, denoted by p(L), is the number of Post complete extensions of L. Thanks to Lindenbaum, we know thatThere is an obvious upper bound, too:Furthermore,.

Sublattices and Initial Segments of the Degrees of Unsolvability

1970 ◽  
Vol 22 (3) ◽  
pp. 569-581 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Lattice Theory ◽  
Initial Segment ◽  
Finite Lattice ◽  
Equivalence Relations ◽  
Finite Lattices ◽  
Image Position ◽  
Degrees Of Unsolvability

In this paper we shall prove that every finite lattice is isomorphic to a sublattice of the degrees of unsolvability, and that every one of a certain class of finite lattices is isomorphic to an initial segment of degrees.Acknowledgment. I am grateful to Ralph McKenzie for his assistance in matters of lattice theory.1. Representation of lattices. The equivalence lattice of the set S consists of all equivalence relations on S, ordered by setting θ ≦ θ’ if for all a and b in S, a θ b ⇒ a θ’ b. The least upper bound and greatest lower bound in are given by the ⋃ and ⋂ operations:

Hamiltonian Circuits on the N-Cube

1975 ◽  
Vol 17 (5) ◽  
pp. 759-761 ◽  
Author(s):  
D. H. Smith
Keyword(s):  
Upper Bound ◽  
Hamiltonian Circuits ◽  
Image Position

The problem of finding bounds for the number h(n) of Hamiltonian circuits on the n-cube has been studied by several authors, (1), (2), (3). The best upper bound known is due to Larman (5) who proved that .In this paper we use a result of Nijenhuis and Wilf (4) on permanents of (0, 1)- matrices to show that for n≥5where τ, a and c are constants.

A Note on the Diameter of a Graph

1968 ◽  
Vol 11 (3) ◽  
pp. 499-501 ◽  
Author(s):  
J. A. Bondy
Keyword(s):  
Shortest Path ◽  
Upper Bound ◽  
Connected Graph ◽  
Maximum Distance ◽  
Image Position

The distance d(x, y) between vertices x, y of a graph G is the length of the shortest path from x to y in G. The diameter δ(G) of G is the maximum distance between any pair of vertices in G. i.e. δ(G) = max max d(x, y). In this note we obtain an upper boundx ε G y ε Gfor δ(G) in terms of the numbers of vertices and edges in G. Using this bound it is then shown that for any complement-connected graph G with N verticeswhere is the complement of G.

scholarly journals On Komlós’ tiling theorem in random graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 113-127
Author(s):  
Rajko Nenadov ◽  
Nemanja Škorić
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Minimum Degree ◽  
Arbitrary Graph ◽  
Spanning Subgraph ◽  
Delta G ◽  
Image Position ◽  
Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

scholarly journals The Maximum of a Random Walk and Its Application to Rectangle Packing

1998 ◽  
Vol 12 (3) ◽  
pp. 373-386 ◽  
Author(s):  
E. G. Coffman ◽  
Philippe Flajolet ◽  
Leopold Flatto ◽  
Micha Hofri
Keyword(s):  
Random Walk ◽  
Upper Bound ◽  
Symmetric Random Walk ◽  
Rectangle Packing ◽  
Unit Width ◽  
Minimum Height ◽  
Image Position

Let S0,…,Sn be a symmetric random walk that starts at the origin (S0 = 0) and takes steps uniformly distributed on [— 1,+1]. We study the large-n behavior of the expected maximum excursion and prove the estimate,where c = 0.297952.... This estimate applies to the problem of packing n rectangles into a unit-width strip; in particular, it makes much more precise the known upper bound on the expected minimum height, O(n½), when the rectangle sides are 2n independent uniform random draws from [0,1].

Injective Hulls of Semilattices

1970 ◽  
Vol 13 (1) ◽  
pp. 115-118 ◽  
Author(s):  
G. Bruns ◽  
H. Lakser
Keyword(s):  
Upper Bound ◽  
Partially Ordered Set ◽  
Binary Operation ◽  
Ordered Set ◽  
Upper Bounds ◽  
Partial Ordering ◽  
Partially Ordered ◽  
Image Position ◽  
Injective Hulls

A (meet-) semilattice is an algebra with one binary operation ∧, which is associative, commutative and idempotent. Throughout this paper we are working in the category of semilattices. All categorical or general algebraic notions are to be understood in this category. In every semilattice S the relationdefines a partial ordering of S. The symbol "∨" denotes least upper bounds under this partial ordering. If it is not clear from the context in which partially ordered set a least upper bound is taken, we add this set as an index to the symbol; for example, ∨AX denotes the least upper bound of X in the partially ordered set A.

Growth conditions on subharmonic functions and resolvents of operators

1985 ◽  
Vol 97 (2) ◽  
pp. 321-324
Author(s):  
J. R. Partington
Keyword(s):  
Banach Space ◽  
Numerical Range ◽  
Bounded Operator ◽  
Complex Banach Space ◽  
Growth Conditions ◽  
Subharmonic Functions ◽  
Image Position

Let X be a complex Banach space and T a bounded operator on X. The numerical range of T is defined by

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