Extremal p-Adic L-Functions

In this note, we propose a new construction of cyclotomic p-adic L-functions that are attached to classical modular cuspidal eigenforms. This allows for us to cover most known cases to date and provides a method which is amenable to generalizations to automorphic forms on arbitrary groups. In the classical setting of GL2 over Q, this allows for us to construct the p-adic L-function in the so far uncovered extremal case, which arises under the unlikely hypothesis that p-th Hecke polynomial has a double root. Although Tate’s conjecture implies that this case should never take place for GL2/Q, the obvious generalization does exist in nature for Hilbert cusp forms over totally real number fields of even degree, and this article proposes a method that should adapt to this setting. We further study the admissibility and the interpolation properties of these extremal p-adic L-functionsLpext(f,s), and relate Lpext(f,s) to the two-variable p-adic L-function interpolating cyclotomic p-adic L-functions along a Coleman family.

New approach to hard corrections in precision QCD for LHC and FCC physics

We present a new approach to the realization of hard fixed-order corrections in predictions for the processes probed in high energy colliding hadron beam devices, with some emphasis on the large hadron collider (LHC) and the future circular collider (FCC) devices. We show that the usual unphysical divergence of such corrections as one approaches the soft limit is removed in our approach, so that we would render the standard results to be closer to the observed exclusive distributions. We use the single [Formula: see text] production and decay to lepton pairs as our prototypical example, but we stress that the approach has general applicability. In this way, we open another part of the way to rigorous baselines for the determination of the theoretical precision tags for LHC physics, with an obvious generalization to the FCC as well.

Secant Spaces to Curves

A classical question in algebraic geometry is whether a given projection of a projective space induces an isomorphism on a given closed subvariety. To answer it, one investigates secant lines to the subvariety. There has been a lot of recent activity in this field ([12], [14],[18], [21], [23]): see [14] and [12] for references).An obvious generalization of the secant lines is provided by the secant r-planes, which intersect a given closed subvariety in r + 1 linearly independent points. The closure of the set of these secant r-planes is the secant variety, and the aim of this paper is to determine its rational equivalence class in the case of curves. There is an extensive classical literature about this problem.

An Expected Utility Theory of International Conflict

An expected utility theory of necessary, but not sufficient, conditions for the initiation and escalation of serious international conflicts, including war, is proposed. The theory leads to the seemingly obvious generalization that actors do not initiate wars—or serious disputes—if they do not expect to gain from doing so. Underlying that generalization are a number of counterintuitive deductions. For instance, I show that though a weak nonaligned state cannot rationally attack a stronger nonaligned nation, it might be able to attack a stronger adversary that, in addition to its own strength, expects to derive support from allies. I also show that serious conflict is more likely between very close allies than between enemies. Systematic tests, using data on serious international threats, military interventions, and interstate wars, as well as 17 cases of known attempts at deterrence, show very substantial support for the expected utility propositions deduced from the theory.

A note on the convergence of multi-point Taylor's series

Series which behave rather like Taylor's series at more than, one point are described and their convergence is discussed briefly.Letwhere z, zs are complex numbers and zs are the points, (the base points)about which a function is to be expanded. They need not be distinct. p{z) is a polynomial of degree n. in z. Assume first that z, zs are real and the zs are distinct. Then polynomials pn-1,t(z) may be chosen so that the serieshas contact of order N at the points zs with any specified real function which has N derivatives at these points. This is proved merely by rearranging the simplest polynomial in z which has contact of order N with/(z) at z = zs. The term given by t = 0 in (2) allows the series to have the values f(zs) at z = zs; adding the term for t = 1 does not alter the values of the series at z = zs; it allows the series to have a derivative with values f′(zs) at z = z; similarly adding the term for t = 2 does not alter the value of the series or its first derivative at z = zs, it allows the series to have a second derivative with values f″{zs) at z = zs; and so on. If m of the zs coincide then f(z) has mN + m− 1 derivatives there and contact is of order mN + m− 1. If z, zs are complex the result is seen to be valid subject to the obvious generalization of ‘ contact of the ith order’ to complex variables and functions.

Which distributive lattices are lattices of closed sets?

1. An elegant theorem due to Tarski states that a completely distributive complete Boolean algebra is isomorphic with a lattice of sets, and in fact the lattice of all the subsets of some aggregate. The obvious generalization of the question underlying this theorem is to ask whether one can pick out by means of a distributivity condition those lattices (not necessarily Boolean algebras) which are isomorphs of lattices of sets. The answer is no. The real numbers with their natural order form a complete lattice which satisfies the strongest possible distributivity conditions and yet is not iso-morphic with any lattice of sets.

Set-Transitive Permutation Groups

The concept of an s-ply transitive (1 ≤ s ≤ n) permutation group on n symbols is of considerable importance in the classical theory of finite permutation groups, which was in the height of its development in the period around the turn of the century. The obvious generalization to a permutation group which is s set-transitive (i.e., a group which, for each pair of s-element unordered subsets S, T of the given n symbols, contains a permutation which carries S into T) seems to have received little attention.

THE INFLUENCE OF SAMPLE COMPOSITION ON MAGNESIUM, CADMIUM, AND LEAD INTENSITY RATIOS AS RADIATED FROM A SPARK SOURCE

Marked variations have been observed in the relative intensities of magnesium, cadmium, and lead lines on the addition of various amounts of foreign substances to standard samples and excitation in a condensed spark source. The changes in the interspectra intensity ratios depended on the nature and amount of the added substances, as well as on the elements under consideration. In many instances buffering of the samples with sodium potassium tartrate or potassium nitrate did not reduce the variations. The behaviour cannot be explained solely on the basis of changes in an "effective" discharge temperature. The data permit no obvious generalization that might be helpful as a criterion for the choice of internal standard elements and buffer substances.