Counting and Probably with Overlap

LastfallatCentralMiddleSchool,43boysand 5 4girlsplayedonasoccerteam .Sincethereisnooverlapbetweenboysandgirls,itiseasytofindhowmanystudentsplayedsoccer .UsethePutting TogetherModelforAdditiontoget 43 + 5 4=97.Youcandrawapicture ,calledaVenndiagram ,toillus tratethissituation ,Separatecirclesrepresentquantitieswithnooverlap .CountingWhenThereIsOverlapSituationswithoverlaparemorecomplicated .Example3 7girlsfromCentralMiddleSchoolplayedbasketballinthewinter .If 9girlsplayedbothsoccerandba…     

Auslander’s defect formula and a commutative triangle in an exact category

We prove Auslander’s defect formula in an exact category,and obtain a commutative triangle involving the Auslander bijections and the generalized Auslander Reiten duality.     

Linear recurrences associated to rays in Pascal’s triangle and combinatorial identities

Our main purpose is to describe the recurrence relation associated to the sum of diagonal elements laying along a finite ray crossing Pascal’s triangle. We precise the generating function of the sequence of described sums. We also answer a question of Horadam posed in his paper [Chebyshev and Pell connections, Fibonacci Quart. 43 (2005), 108–121]. Further, using Morgan-Voyce sequence, we establish the nice identity $F_{n + 1} - iF_n = i^n \sum\limits_{k = 0}^n {(_{2k}^{n + k} )( - 2 - i)^k } $ of Fibonacci numbers, where i is the imaginary unit. Finally, connections to continued fractions, bivariate polynomials and finite differences are given.     

Modules with Nakayama’s property

Modules M _A with Nakayama’s property are studied. In particular, for a right invariant ring A, it is proved that all right A-modules satisfy Nakayama’s property if and only if the ring A is right perfect.     

On the Kernel of the Borel’s Characteristic Map of Lie Groups

For compact and connected Lie group $G$ with a maximal torus $T$ the quotient space $G/T$ is canonically a smooth projective manifold, known as the complete flag manifold of the group $G.$ The cohomology ring map $c^∗: H^∗ (B_T) → H^∗ (G/T)$ induced by the inclusion $c:G/T→B_T$ is called the Borel’s characteristic map of the group $G [7, 8],$ where $B_T$ denotes the classifying space of $T.$ Let $G$ be simply-connected and simple. Based on the Schubert presentation of the cohomology $H^∗ (G/T)$ of the flag manifold $G/T$ obtained in $[10, 11],$ we develop a method to find a basic set of explicit generators for the kernel ker$c^∗ ⊂ H^∗ (B_T)$ of the characteristic map $c.$     

Chen’s Theorem with Small Primes

Let N be a sufficiently large even integer. Let p denote a prime and P2 denote an almost prime with at most two prime factors. In this paper, it is proved that the equation N - p+ P2 (p ≤ AT0.945) is solvable.     

The Bertrand’s manifolds with equators

Natural mechanical systems describing the motion of a particle on a two-dimensional Riemannian manifold of revolution in the field of a central smooth potential are studied. A complete classification of such Riemannian manifolds and potentials on them satisfying the strengthened Bertrand property, i.e., any orbit not contained in any meridian is closed, is obtained.     

Waring’s problem with digital restrictions

The aim of this paper is to consider an analogue of Waring’s problem with digital restrictions. In particular, we prove the following result. Lets _q (n) be theq-adic sum of digits function and leth,m be fixed positive integers. Then fors>2 ^k there existsn ₀∈ℕ such that each integern≥n ₀ has a representation of the form We will even give an asymptotic formula for the number of representations ofn in this way. The result is shown with help of the circle method in combination with a “digital” version of Weyl’s Lemma. Dedicated to Professor Hillel Furstenberg The first author was supported by the Austrian Science Foundation project S8310. The second author was supported by the Austrian Science Foundation project S8308.     

Resolving singularities with Cartan’s prolongation

A standard method for resolving a plane curve singularity is the method of blow-up. We describe a less-known alternative method which we call prolongation, in honor of Cartan’s work in this direction. This method is known to algebraic geometers as Nash blow-up. With each application of prolongation the dimension of the ambient space containing the new “prolonged” singularity increases by one. The new singularity is tangent to a canonical plane field on the ambient space. Our main result asserts that the two methods, blow-up and prolongation, yield the same resolution for unibranched singularities. The primary difficulties encountered are around understanding the prolongation analogues of the exceptional divisors from blow-up. These analogues are called critical curves. Most of the critical curves are abnormal extremals in the sense of optimal control theory as it applies to rank 2 distributions (2 controls). Dedicated to V. I. Arnol’d and his creative force     

Sturm-Liouville’s problem with discontinuous nonlinearity

We establish a theorem on the existence of solutions of the Sturm-Liouville problem with nonlinearity discontinuous in the phase variable. By way of application, we consider a one-dimensional analog of the Gol’dshtik model of separated flows of an incompressible fluid.     

Baker’s conjecture and Eremenko’s conjecture for functions with negative zeros

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metric, together with new results which show that the images of certain curves must wind many times round the origin.     

Revisiting key schedule’s diffusion in relation with round function’s diffusion

We study the weakness of key schedules from an observation: many existing attacks use the fact that the key schedules poorly distribute key bits in the diffusion path of round functions. This reminds us of the importance of the diffusion’s relation between key schedules and round functions. We present new cryptanalysis results by exploring such diffusion relation and propose a new criterion for necessary key schedule diffusion. We discuss potential attacks and summarize the causes for key schedules without satisfying this criterion. One major cause is that overlapping between the diffusion of key schedules and round functions leads to information leakage of key bits. Finally, a measure to estimate our criterion for recursive key schedules is presented. Today designing key schedule still lacks practical and necessary principles. For a practical key schedule with limited diffusion, our work adds more insight to its requirements and helps to maximize the security level.     

Refined Young inequalities with Specht’s ratio

In this paper, we show that the ν-weighted arithmetic mean is greater than the product of the ν-weighted geometric mean and Specht’s ratio. As a corollary, we also show that the ν-weighted geometric mean is greater than the product of the ν-weighted harmonic mean and Specht’s ratio. These results give the improvements for the classical Young inequalities, since Specht’s ratio is generally greater than 1. In addition, we give an operator inequality for positive operators, applying our refined Young inequality.     

Second Noether’s theorem with time delay

In this paper, we consider variational problems with delayed argument. Simple proofs of the first and second Noether theorems are given. Applying our results to some concrete problems we obtain conservation laws for systems with delay.     

Planar Lattice Graphs with Gallai’s Property

In 1966 T. Gallai asked whether connected graphs with empty intersection of their longest paths do or do not exist. After examples of such graphs were found, the question was extended to graphs of higher connectivity, and to cycles instead of paths. Examples being again found, for connectivity up to 3, the question has been asked whether there exist large families of graphs without Gallai’s property. The family of grid graphs, a special kind of graphs embedded in the planar square lattice ${\mathcal {L}}$ , has been shown by B. Menke to contain no graph enjoying Gallai’s property. In this paper we find several examples of graphs embedded in ${\mathcal {L}}$ and enjoying that property with respect to both paths and cycles.     

Amply regular graphs with Hoffman’s condition

It is known that, if the minimal eigenvalue of a graph is −2, then the graph satisfies Hoffman’s condition: for any generated complete bipartite subgraph K _1,3 (a 3-claw) with parts {p} and {q ₁, q ₂, q ₃}, any vertex distinct from p and adjacent to the vertices q ₁ and q ₂ is adjacent to p but not adjacent to q ₃. We prove the converse statement for amply regular graphs containing a 3-claw and satisfying the condition μ > 1.